Corrosion in the cable or connection... easiest way to find where (if it
isn't obvious on a visual inspection) is to perform a voltage drop test. If
you have a multimeter, the instruction booklet should describe how to
perform this test.

Common problem, what happens is the acid eats up into the cable under the
insulation. What you really have is just a few good strands of copper still
in place. When you hit the key the remaining strands get cherry red hot.
Simply replace the neg cable from the battery to the engine block. I suspect
the other cable might not be much better condition.
You can get them pre-sized at many auto parts stores with the fittings
already installed on the ends.
Let us know if this helps...