This idiom means to contact somebody briefly. The most common reasons for wanting to 'touch base' with someone are to find out how they are doing, what they think about something, or to update them or receive an update about something. It can also be used to refer to renewing contact with a person you have not communicated with for a while, like an old friend.
As the video shows, this expression can be used in a social context but is more common in a professional one, particularly in the corporate world where it has become increasingly popular. In a busy workday, you usually would not have time for lengthy conversations. Therefore, it is preferable to simply 'touch base' with someone to find out what is happening or how something is developing.
Phrases with a similar meaning include 'drop a line,' 'check in with,' and 'reach out to.'