Out of universe (aka, in the real world), OPM is definitely a parody of seinen and superhero stories (Power Rangers type stuff, in particular), especially at its inception. So his power simply is, and serves to parody the aforementioned genres, as indicated in HamtaroWarrior's answer. He is just the hero that shows up to vanquish the monster at the very end of the story, only in this case he doesn't do it for Love or Justice or any high ideals like that.
In-universe, though, there have been certain indicators that Saitama's power was obtained by constantly ignoring his limits until eventually he stopped having any. See Chapter 56 of the webcomic*, in particular:
The following is Dr. Genus speaking to Zombieman about why he has given up on the House of Evolution--Saitama made him realize it was futile, as all he would ever do was raise the limits of humanity, which could never compete with something without limits.
There are a few panels before and after this one that continue to discuss Saitama's power. Go read the rest of the chapter for them.
The manga is not quite this far along yet, but it's not terribly far away, and there are some things in the manga that are not in the webcomic (see the image and surround text below, for example).
Someone who is already in decent shape will not be fazed much by Saitama's exercise routine (see HamtaroWarrior's answer). Their biggest problem, if any, would be the lack of rest days. However, people in bad shape would take to it much differently. I know in my current couch-potato state I'd have a lot of trouble with jumping straight into that kind of routine, and would likely inflict several long-term injuries on myself from muscle pulls etc. I've basically done so in the past, actually. Years ago, when I was a scrawny little nerd who never really exercised, I suddenly decided to try to get some muscles and tried getting 100 push-ups done total throughout the course of a day (2 here, 10 there, etc.). I pulled a stabilizing muscle around the spine and shoulder blade. I collapsed face-first onto the floor with a scream when it happened. For a few days I had limited use of one of my arms and spent most of my time in bed (which was still painful--you never really stop trying to use a stabilizing muscle, thanks to gravity), and I had painful muscle spasms in that area for many years.
Saitama was probably in better shape than that when he started, but he wasn't in particularly great shape either. He was just a lazy, unemployed lay-about, after all. I would expect him to be pretty firmly on the side of "doing this every day will lead to injuries and a lot of pain" when he started training.
The manga adaptation has a sidestory about Saitama's training days that shows him powering himself past intense pain, actually.
It's worth noting that the pain was from a bad tooth, not his training. He was already in his 300th day of training during this story. Once the tooth was knocked out he felt fine, but before that he's in great pain and continues his workout and saves a few people along the way regardless. There's another sidestory early on about his training days, but doesn't go into what pain he may have experienced in the early stages. The anime adaptation of the scene from Hamtaro's answer includes a brief clip showing Saitama in pain while training, though.
Saitama essentially always trained beyond whatever his limits were--if he suffered an injury like I did, he would havewilled himself past it and continued to do 100 pushups and everything else every single day regardless. For the record, this is a really bad idea in real life. Please don't do it. Exercise programs and such should be taken seriously when they tell you to consult with a doctor before starting any exercise routine.
*Or you can now see chapter 88 of the manga ("88th Punch: Limiter") for essentially the same exposition. Be careful if you go looking for this online, though. This series is notorious for having completely different numberings according to different people/sites, to the extent that some sites still have certain earlier chapters out of order. I see this chapter in question listed as chapter 135 in one location, for example.