This is a Nurikabe puzzle. The goal is to paint some cells black so that the resulting grid satisfies the rules1 of Nurikabe:
- Numbered cells are white.
- White cells are divided into regions, all of which contain exactly one number. The number indicates how many white cells there are in that region.
- Regions of white cells cannot be adjacent to one another, but they may touch at a corner.
- Black cells must all be orthogonally connected.
- There are no groups of black cells that form a $2\times2$ square anywhere in the grid.
1 Paraphrased from the original rules on Nikoli
First, any two numbered cells that are one space apart must be separated by a black cell, which gives us the top-right, and given the "all black cells must be connected" condition one straightforwardly (and stair-wise) gets to
Again, from here the continuation still follows from the same rule applied to the bottom-right, until
Now note that the orange square below cannot be white, it's too far away from any numbered cell, and therefore the red one nearby cannot be black, because of the 2x2 rule. Moreover, there is no way the red cell can be connected to another 6 than the one on its right, because the one on its right would not have 6 friends.
We therefore get this, right after a second application of the 2x2 rule:
From here, there is only one way to connect the black pieces while leaving enough space for the top 6 to have friends: