What's the meaning of "get to do something"?
I get to spend a lot of time with you and the kids.
To "get to do" something means that you are enabled or given the opportunity to do something.
Example 1 - A circumstance or choice enables you to do something:
Why do you work at home?
I get to spend a lot of time with you and the kids.
Example 2 - Being given an opportunity:
Has anything good come out of this week-long snowstorm?
I get to spend a lot of time with you and the kids.
Sometimes this phrase is used sarcastically to mean "I am being forced to do something that I don't want to do". For example:
I get to grade 40 tests this weekend!
should be interpreted as something along the lines of
I'd rather be doing something else, but I must grade 40 tests this weekend.
In this context it means have the opportunity to:
I have the opportunity to spend a lot of time with you and the kids.
http://dictionary.reference.com/browse/get
21: to succeed, become enabled, or be permitted: You get to meet a lot of interesting people.
To 'get to do something' means that you are given a chance/opportunity to do that thing.
There are bunch of replacements for it such as to be given a green light, be enabled to, be opened doors to doing.