This phrase suggests a slow, gradual process, not something that happens quickly. He didn't get the role immediately but wormed his way into the cast over several months.
Pro Tip 2/3
Stealth Involved
To 'worm your way into' implies moving into something unnoticed, like a worm burrows unseen. Alex wormed his way into the exclusive club by befriending members one by one.
Pro Tip 3/3
Sometimes Negative
The phrase can have a negative connotation, implying manipulation or deceit. She didn't trust him, thinking he had wormed his way into the position.