O Our Lord! Cover (Us) with Your forgiveness: me, my parents, and (all) believers, on the day that the reckoning will be established! (14:40-41)
One of the first du'aat I learned to make in my salah was one from the Qur'an, a du'a of Ibraheem. In it, Ibraheem asks Allah to make him someone who establishes prayer--although the translation I learned inserted the word "regular," i.e., "establishes regular prayer." This du'a reminds me, at the end of every salah, the important of salah, of establishing it and praying it regularly.
On just about every prayer timetable I've seen, part of an ayah is listed somewhere on the page. One translation of the part of the ayah, 4:103, is "Verily, the prayer is enjoined on the believers at fixed hours." The idea is to remind whoever reads that prayer table about the importance of praying regularly at the appropriate times.
Now a person can view the idea of regular prayer as either a burden, or a blessing. I have a hunch most non-Muslims, and plenty of Muslims, probably see it as a burden. And undoubtedly Shaytan would rather us see it as a burden, so he can easily distract us from it, urge us to procrastinate it, and eventually even convince us to abandon it altogether. May Allah protect the believers from his whispers.
There are benefits in having the prayers spread throughout the day. It gives you a spiritual retreat at key points during the day, to help you break up the day and refresh you. And the times of the salah are intricately connected with ideal daily behavior.
We learn the prayer times from the Prophet Muhammad ﷺ, who learned them from the Angel Jibreel over two days. According to Ibn Abbas, the Angel Jibreel visited the Prophet Muhammad ﷺ at the beginning of the each of the prayer times on the first day to lead him in prayer, and on the second day led him in prayer at the end of the prayer times, except for maghrib prayer. The times have been further specified by Abdullah bin 'Amr bin al-'As, based on the sun and sky, and scholars have differed slightly in their opinions as to the exact timing.
What's clear however is that the prayers are based on the timing of the sun, indicating that our days should follow a similar schedule. It also keeps us Muslims aware of the motion of the sun throughout the day, as it crosses the sky, and throughout the year as the time it takes to traverse the sky changes. In this way the timings of prayers keeps you alert, and it keeps you from forming a lazy habit or tradition when it comes to the prayer--your schedule will have to be flexible somewhat throughout the year. The beginning and end times for each prayer vary between some schools of thought, though not drastically so and not without evidence.
The first prayer of a waking day is fajr, and there is unanimous agreement regarding its start and end times. It begins at the time of the "second dawn" or "true dawn." While the sun is at one particular angle below the horizon, there will appear the "first dawn" known as the "false dawn," when the light spreads vertically. That is not the start of fajr time, which actually comes later, when the sun is high enough for the dawn light to spread laterally across the horizon. It ends when the sun rises. This means that our day should begin before the sun comes up. There's also a special blessing in the fajr time before the sun rises. While our minds and bodies are refreshed, it can be a very productive time of day before the worries and business of the day start to clog our minds.
The start time of dhuhr is also unanimously agreed upon--that it is when the sun declines from its zenith. Geographically, unless a person is at the equator he will have a small amount of shadow, even when the sun is at its zenith, but the zenith is when the shadow has reached its minimum size. There are two opinions about the end time of dhuhr, though they all agree that dhuhr ends at the time when asr begins. The first opinion, the Maliki, Shafi'i, and Hanbali opinion, is that dhuhr ends when the length of an object's shadow is equal to its height (plus the "extra shadow" just mentioned.) The second opinion, the Hanafi opinion, is that dhuhr ends when the length of an object's shadow is twice its height (plus the "extra shadow.") This is based on a hadith that dhuhr is to be delayed on hot days until the day begins to cool off.
The start time of asr is agreed by all to be the end time of dhuhr, and the same differences just mentioned apply. There is also agreement as to the end time of asr, that it be when the sun has completely set. Scholars also agree that it is better to pray asr earlier (than later) as long as its in the specified time. Hanafi scholars prefer it to be delayed as long as the sun hasn't started to change color.
By unanimous agreement, maghrib time begins when the sun has set, though there are basically three opinions regarding its end time. The first is the Maliki and new Shafi'i opinion, that basically the time for maghrib ends once enough time has passed to actually make wudhu, adhan, iqama, and pray five raka'at (3 for fard, 2 for sunnah.) In other words, maghrib needs to be prayed right away with no "extended time." The Hanbali and old Shafi'i opinion is that maghrib needs to be prayed by the time the red twilight has faded, while the Hanafi opinion is that it may be prayed until the white twilight has faded. But they all pretty much agree that it's best to pray maghrib at the beginning of its time.
When it comes to isha, there is unanimous agreement that it begins when the twilight has faded, but there are the same differing opinions about which twilight that means. The Maliki and Shafi'i opinions, for which there is no extended time, also say isha starts after the twilight has faded. When the sun sets, the first twilight is the red twilight, followed by the white twilight, followed by the blue twilight, just as a point of reference. There are two opinions about the end time of isha. The first is the Hanafi opinion, which allows for isha to be prayed up until the time for fajr arrives. The Maliki, Shafi'i, and Hanbali opinions are that isha may be prayed until the end of the first half, or first third of the night. This is calculated as the time between the beginning of isha and the beginning of fajr, then split into thirds or halves and added to the time isha begins.
The salah itself is an organizational tool, to help you structure your life. Sometimes people (you know who you are) will say that time is money. But no, time is life. Whenever a day passes, part of you goes with it. Following the salah forces you to begin your working day with fajr time--you shouldn't go to bed after salat al-fajr. You also see that there is time to take a break, for dhuhr, a good time to eat lunch, and maybe take a nap. Asr time, when the day starts to draw to a close, is the time to stop working and see to your family. Eat dinner and prepare for bed, these are things to do in the evenings.
Even the prohibited times of prayer reminds us of the appropriate structure for the day, so we don't turn into monks and try to pray the entire day--there are times that we should spend doing other things as well. But the larger point of regular prayer is to prevent other things, our life in this dunya, from stunting our relationship with Allah.