What is connect?

connect is a method of ROBLOX Events. Methods use : instead of . before their name (use :connect not .connect)

We connect to an event (like Touched, the event on a part that fires when something touches it; or PlayerAdded the event of the Players service which fires when a new player joins). Wiki section on connecting

The connection asks for a function value which is called by the connection -- we tell it what to do when the event occurs.

As a simple example, here's a simple script utilizing connections.

-- A function which destroys whatever you give it
function DestroyPart(part)
    part:Destroy();
end

script.Parent.Touched:connect( DestroyPart );
-- Connect to `script.Parent` `Touched` event (assuming it's a Part)
-- telling it that we want to destroy anything that touches it

Here is a version of that script broken down a bit more:

local DestroyPart = function(part)
    part:Destroy();
end
-- `DestroyPart` is actually a VARIABLE
-- that we can pass as an argument when
-- `connect`ing

local touchEvent = script.Parent.Touched;

-- Ask to `connect` to `touchEvent`
touchEvent:connect(DestroyPart);
-- telling it to use the function that's in the variable `DestroyPart`