Let us see if any help we can get from its ancient scripts: (image taken from http://vividict.com)
|Evolution history of 解|
Based on its original meaning, 解 is extended to mean dissect, break up or cut up. For example 解剖.
Since 解 contains the meaning of "to divide and separate parts of a buffalo", 解 is also used to mean untie. 解开.
Once the buffalo is finally dissected, it means the task is accomplished, 解 thus has meaning of resolve. for example 解决.