How does a restriction enzyme know where and what to cut in DNA?

The enzyme feels its way along the DNA, and cuts where the shape fits best. (A, G, C, and T each have their own shape.)

The restriction enzyme simply cuts at a certain nucleotide sequence, which corresponds with the general locale of the gene in question.  The enzyme does not 'know'.

You also mentioned '...and paste' in your query.  Restriction enzymes don't do the 'pasting', just the cutting.

Great questions - keep studying science!

Restriction Enzymes:
These special enzymes were discovered in the late 1960's as naturally occurring agents in bacteria. They protect the bacterium against foreign DNA from other organisms. Invading DNA is cut into pieces and made inoperable. This process is called restriction. As with any enzyme, these are specific in the job they do. Many of them only recognize short, specific nucleotide sequences( recognition sequences) and cut at specific points within those sequences. Bacteria protect their own DNA by a process called methylation. During this process methyl groups are added to the nucleotides within the recognition sequence. There are several hundred restriction enzymes and about 100 different recognition sequences.
Recognition Sequences: are symmetric in that the same sequence of 4 to 8 nucleotides is found on both strands, but run in opposite directions. The restriction enzymes usually cut the phosphodiester bonds of both strands in a staggered manner. The result being both ends have a single stranded area called the sticky ends. It is within this space that the new piece of DNA is added, attaching to the sticky ends.
These unions are temporary until the enzyme DNA ligase is added to catalyze the formation of the phosphodiester bonds.
Vectors: are used as carriers for moving DNA from test tubes into cells. Bacterial plasmids and viruses are the most widely used vectors in DNA transfer. Bacterial cells can pick up the DNA through the process of transformation. Lambda phages are used by eliminating the middle of its liner genome and adding the foreign DNA in the created space. The phage is then introduced into the bacterial cell where it replicated itself via the lysogenic cycle.

Useful REs bind to very specific sequances, but others are less specific (and even useful ones can be less specific - star activity - when put in sub optimal conditions). It is my understanding that enzymes recognize their targets much like any other enzyme... through hydrogen bonds, electrostatic interactions, "hydrophobic forces" and Van der Waal forces.

The double helix is not a homogeneuos structure, rather there are bends which the RE can recognize and deviations in DNAs' structure from the norm  (this is probably even more important for RNAs). Furthermore, the four differant nucleotides can interact differantly with the amino acids in the catalytic sites of REs through the major groove in the double helix. That is where the REs can make important hydrogen bonds with some amino acids (and not others) that stabilize the interaction between the RE and the DNA.