To “click with” another person is to have a strong, effortless personal connection with them. If you meet someone and talk for an hour without any awkwardness, you probably 'click' well with this person. For a romantic relationship, this “click” is especially important:

I didn’t really click with anyone I met on any of those online dating sites.

We probably wouldn’t use this term in formal or business writing, but it’s quite common in speech and casual written communication.