Imagine you want to tour the 10 biggest cities in the UK, starting in London (number 1) and finishing in Bristol (number 10). The solution to the Hamiltonian Path Problem is the the shortest possible route you can take.
This simple problem is surprisingly difficult to solve. There are over 3.5 million possible routes to choose from, and a regular computer must try them out one at a time to find the shortest. Alternatively, a computer made from millions of bacteria can look at every route simultaneously. The biological world also has other advantages. As time goes by, a bacterial computer will actually increase in power as the bacteria reproduce.
These developments in synthetic biology are really amazing: it is just another example of how researchers are looking at pre-existing biological structures to solve problems (albeit somewhat abstract problems in this case) instead of building technologies from scratch.