How is the Longest Path Problem NP complete?
How does that guarantee that there is a hamiltonian path from s-t?


It doesn't, but the Longest Path problem doesn't require a path between particular vertices, it only requires that some simple path of length $K$ exist in the graph.  To reduce Hamiltonian Path to Longest Path you just require that path to have $|V| - 1$ edges, which in a simple path must involve all the vertices in the graph, making it a Hamiltonian Path.