- Wittgenstein's Picture Theory of Meaning was an attempt to explain how language represents the world. His suggestion was pretty much that it works by being isomorphic to the world, in some way or another: parts of sentences map onto parts of the world, and relations between those parts map onto relations between parts of the world. (It's plausible that this is how pictures represent. Suppose I have an illustration of the solar system. This blue blob maps corresponds to Earth, this red blob corresponds to Mars, this big yellow blog corresponds to the Sun, etc.) It's intended, I think, as a metaphysical theory of how representations in general are possible, including language.
- Fodor's Language of Thought Hypothesis is an attempt to describe how human thought is structured. His suggestion is that it is specifically language-like. That is, it is language-like even in ways that pictures and other kinds of representation aren't.
I'll try to get at what's specific to the LOT hypothesis. Fodor's own explanation can be found in his essay Why There Still Has to be a Language of Thought.
Fodor is an intentional realist. Intentional realists think that intentional states ("believes that __", "hopes that __", and whatever else) literally and concretely exist. For someone to be in a particular intentional state, is to have a literal thing in their head that stands for a proposition, and for them to be in an appropriate relation to that thing.
Fodor shares a colourful metaphor that he got from Stephen Schiffer. Inside my head there is a thing, let's say it's a pebble, which somehow has come to represent the proposition that "The soup has gotten cold". I also have a bunch of boxes in my head, which the pebble can go in. If I believe that the soup has gotten cold, then the pebble is in my "belief box". If I hope that the soup has gotten cold, then the pebble is in my "hope box". If I fear that... etc.
(And if I believe that the dog bit the man, then a different pebble is in my belief box.)
So far that's just intentional realism. The LOT hypothesis is specific about what the things, the representations, have to be like. In LOT, they can't be pebbles.
If LOT is false, it might be the case that there is a representation for "Bob loves Jill", and a different representation for "Jill loves Bob", but also for the two representations to have nothing to do with each other. They might be two pebbles (no more similar to each other than either one is to the pebble representing "It was the best of times, it was the worst of times..."). It might be possible to have an animal capable of having the thought that Bob loves Jill, but incapable of having the thought that Jill loves Bob (since it has one of the pebbles but not the other one).
LOT says that representations of propositions have constituent structure. That is, the "Bob loves Jill" representation is made of parts; and furthermore, those are the same parts that the representation "Jill loves Bob" is made of. That's why it's absurd to imagine someone understanding the first sentence but not the second. If you have the constituents, then you can understand strings which are put together out of those constituents.