How did 涉 semantically shift to mean 'to experience, be involved'?
Great question. The original meaning of "涉" is that stepping into and wading across the stream by my feet. (The phrases "涉水"、"涉江" are specific examples of the original meaning.)  

In my opinion, "涉", to some extent, can be regarded as "step into".(Not very exact) 

And if I stepped into something dangerous yesterday, it also means that I experienced something dangerous yesterday. (The phrase "涉猎" is the specific example of the second meaning. )

And if something "stepped into" me, it also means that I was involved in something. ("涉及"、 "涉外". To some extent, you can understand "涉" as "about". )

Examples：

1、我一路跋山涉水。
explanation: I've been trekking all the way. (跋山涉水: trekking. And "涉" means that I wade across the rivers.)

2、涉险事故。
explanation：Misadventure. (The accident("事故" means something bad) about("涉") dangerousness(险))

3、他涉猎广泛。
explanation：He has a wide range of interests. (He step into("涉") and hunt("猎") a lot of things.)