Yesterday I bought my first DSLR, a used Canon EOS Rebel T5 with a lot if accessories. It has been used 15 times, and I think I’m gonna take a lot of pictures with it, but what happens when I take too many pictures? Will the shutter stop working? Could I fix it?
The camera shutter is a mechanical part, and like all things in the universe, subject to eventual failure. You can see How many actuations are "too many actuations"? for some information on just how many clicks a typical shutter mechanism is good for (depending on the model, between 50,000 and 400,000). But note that these are just estimates based on averages — it's perfectly normal for a shutter to develop a problem much earlier, or to last much longer.
As for what happens when it dies, well: How much does it cost to replace a DSLR shutter? In the case of a used older-model camera, probably more than you spent in the first place. This is generally not a do-it-yourself repair.
But, I encourage you to not worry about this too much, unless your intention is to take many time-lapse sequences or some other special case. If you're just shooting, you should consider this getting value out of your purchase. If the camera is just sitting there and you're not using it for fear of wearing it out, then your purchase would be really going to waste.
A count of just 15 seems unlikely on a used camera!
No, it won't turn off at some predetermined number. It's just, like an odometer on a car, an indicator of how much it's been used. Interesting when determining the resale price perhaps.
But no, shutter replacement if it DOES fail is not a DIY job, and is very likely not economic.