It has been some time since I finished the second season now, but I will try to go on what I remember:
There were several reason why Walker was in a difficult position and why his opinion ratings were dropping quickly.
- Walker (supposedly) led the country into a growing conflict with China, which possibly could have been avoided. This also led to an energy crisis in the US with raising energy prices. All this was not exactly helping his reputation. Of course, Frank orchestrated all of this behind the curtains.
- On top of all that, there was a major investigation going on about illegal donations for his party over the past years. This was bad press and Walker had to cut ties with his biggest ally, Tusk, because of this.
- Walker and his wife had marital problems. Frank and Claire convinced them that they should seek counseling, which they did. When it finally got uncovered, it was also unveiled that the therapist gave Walker drugs with psychological effects. This was a huge scandal, because it now was suggested that Walker's judgement was impaired. Especially with the current crisis going on, this meant a huge lost of trust for him.
- In consequence, his polls were dropping rapidly and it seemed clear that he had lost a lot of public faith. This was a serious threat for the next elections and could have cost his party the win over the Republicans.
In the end, Walker was isolated, without political allies or public support. Because of the allegations against him, he would probably have faced an impeachment trial and might have lost the office anyway. Resigning was the logical consequence for him.