FPGA: How do LUT's change their logic
A LUT (Lookup Table) in modern FPGAs is nothing more than a RAM. The inputs are the address lines, and the output is the data output bus. There's really nothing more to it.

FPGAs also tend to have more advanced logic modules (some vendors call them ALMs) which consist of one or more LUTs along with additional dedicated adders, high speed carry chains, and typically an optional register (flip-flop) on the output. The inputs connect to the logic routing resources in the FPGA (which are also typically just RAMs) and in some cases also have a feedback path to connect to the output register of the same ALM.

All of the LUTs are initialised at power on by a configuration file which essentially uses the write port of the LUT RAM to program it. Once configured the memory acts to perform whatever logic function it is programmed to do.

As @duskwuff reminded me in the comments, it is in many FPGAs possible to use the LUT as a RAM directly. In designs where this is the case, additional inputs to each LUT are made available which form the a write data port (typically this port is a single write data bit and a write enable bit, and the address is shared with the read port).

For a diagram of what the ALMs or ALUTs or LABs (or whatever a vendor calls them) look like for a given FPGA, you can typically find this in the documentation for the device family. Different FPGA families will have different structures.

As an example, here is the structure of an ALM (Adaptive Logic Module) for an Altera Cyclone V FPGA (taken from the datasheet on p20). I've added some annotations.



The LUT is on the right, then there are some dedicated full-adder (FA) blocks which allow high-speed multi-bit adders to be made by chaining multiple ALMs together (using the dedicated carry chain routing). Then a multiplexer to select whether or not the FA should be used, and finally the output registers.

There are lots of other signals like clocks and resets for the registers and the configuration interface for the LUT which are usually not shown on the diagrams for simplicity, but they are there.