"Lassen" in this sentence is a bit like an antonyme to "tun". It means something along the lines of "to not do something", "to refrain from doing something" or "to let something happen (without doing something about it)".

To get a better idea, you might want to look at the related verb "zulassen". This is mostly translated along the lines of "to allow", "to permit". But "zulassen" has a more passive vibe than those. You can imagine it as something like "to tolerate something", "to not do anything to prevent something".

You could use "lassen" for example like this:

Ich könnte heute abend Joggen gehen. Oder ich könnte es lassen.

I could go running tonight. Or I could not.

or

Die Nachbarskinder machen wieder Krach im Garten! - Ach, lass' sie doch!

The neighbor's kids are making a ruckus again in the garden! - Oh, leave them be!

The set phrase

Ich kann tun und lassen, was ich will

is indeed a stronger emphasis compared to only "ich kann tun, was ich will". The speaker not only says that they can do whatever they want - it's also in their power to not do something, to refrain from something, or to not do anything at all.

There's also the variation "etwas sein lassen", which can be quite literally translated as "to leave something be".