The clique function takes a bit string representation of a graph with m nodes (one bit for each pair of nodes, 1s where there is an edge between two nodes) and outputs whether the graph has a fully connected subgraph of size s (a clique).
CNF and DNF are conjunctive and disjunctive normal forms, respectively. CNF has the form (x OR (NOT y) OR ... ) AND ((NOT z) OR y OR ... ) AND ..., while DNF has AND and OR exchanged. Any boolean function can be expressed as CNF or DNF, but this might blow up its size exponentially.
The monotone network complexity is the number of binary {AND,OR} gates you need to compute a function. Monotone because increasing the input (setting a bit to 1) never decreases the output. The basic gates have this property, and if you never use NOT the composition has it too. This means that monotone networks can only compute monotone functions. The clique function is monotone, since adding an edge can never destroy an existing clique.
The CNF-DNF-approximators mentioned are a technique for creating a CNF (or DNF) of such a network by introducing a limited amount of errors (hence approximator) at each step. This is done by switching between CNF and DNF at each gate, but discarding parts of the formula that get too large. (I don't really understand how that keeps the error bounded.)
Using the properties of the clique function, it is possible to show that the total number of errors by a limited CNF-approximator must be large, which means that the switching procedure must have been applied many times. This gives a lower bound on the number of gates in any monotone network that computes the clique function, and this bound is exponential.
The paper under discussion attempts to extend this result to non-monotone networks, which can also make use of negation. To do that, it extends the CNF/DNF-switching to also handle negated variables without introducing significantly more errors. (Again, I don't understand how that works.)
Assuming the extension is correct, any bound on the monotone network complexity using CNF-DNF-approximators also holds for the non-monotone network complexity of the given monotone function.
Applying this to the exponential lower bound of the clique function, this means that there is no non-monotone network of a polynomial number of gates that computes it, which implies that there is no polynomial-time Turing machine, which implies P != NP.