Unlike what others have said, using the auto-router isn't the problem. They are right in that you can't just throw a whole design at the auto-router and expect it to solve everything for you. But, when used properly, auto-routers are legitimate and time-saving tools. Don't listen to the knee-jerkers that say not to use the auto-router.
Your problem is that you tried to cram way too much stuff onto a 2 layer board. Expecting to route that many pins that closely spaced in 2 layers is wildly unrealistic.
The other issue is that you didn't consider layout carefully enough. This is harder to evaluate by looking at your images, but it seems quite likely.
For one thing, there is very little room around the dense chip. Even with multi-layers, there will be congestion around that chip. Sometime I even hand-route traces away from a dense chip to expand them out a bit, then see how the auto-router can handle it.
However, the first rule of good routing is good layout. You can't just plunk down parts somewhere, then connect them up somehow in routing later. Good layout is something you'll learn and get some intuition for as you do more designs. For the first few designs, it helps to give yourself lots of room. You haven't.
Large parts are often flexible in their pin assignments. This usually true of microcontrollers and FPGAs. In some cases I've actually printed out a pinout picture of a large part. I then made notes around it corresponding to the rough placement of things it had to connect to on the board. I crossed off all the fixed pins, like power, ground, MCLR, etc. Then I carefully assigned the soft pins based on proximity to the thing they had to connect to.
This can be a iterative process. You may get partway around the part and realize you are one pin short in one direction. That may require re-assigning pins on the other side of the part to shift things around.
For large parts like microncontrollers, I place it in a large empty area, then place only its immediately connected parts around it. This includes the bypass caps, and the crystal with its caps, if any. You then orient and move that whole group of parts together as a unit from then on.
It's perfectly normal to place some parts in only rough positions, then come back and pack them more efficiently as more parts are placed. Again, the whole process is iterative. After you've gotten some experience and intuition, these steps will go quicker. Expect the first few designs, especially dense ones, to take a while.
Once you have a reasonable layout with air wires that don't cross all over the place, do a little manual routing of important signals. I usually do all the bypass caps first, which of course should already be close to the power and ground pins they are bypassing. If you have a ground plane, then the next step is to connect most of the ground points to the ground plane with vias. This leaves only the air wires that will be actual routable traces.
At this point, depending on your experience, you route a few things you can see will be issues, or just let the auto-router fly.
However, you are not using the auto-router yet to create the final route, just to show you the problem spots. Good auto-routing is also a iterative process. You run the auto-router, see where it gets into trouble, do some manual routing and maybe placement changes as a result, run the auto-router again, etc. Eventually you converge on a completed route. The auto-router has still saved you significant time by doing much of the grunt work for you.
After you have a solution you are reasonably comfortable with, you look at everything carefully and manually clean up obvious things. For example, if you have a ground plane, you want to not have vias clumped. Lots of small islands are better than a few larger islands in the ground plane.
Again though, don't listen to all the religious knee-jerkers. Go ahead and use the auto-router, but do it carefully and responsibly. I do electrical engineering professionally, and have used the auto-router in some way on probably over 95% of all boards I've designed. The more complicated the board, the more the auto-router is a valuable tool in doing the grunt work for you. Just don't expect it to ever do all the work. And, you have to start with good placement.