The AEC website you linked to contains pretty much all the relevant information.
The Senate only appears to have more candidates, because of the way Senate seats are structured. When voting for the House of Representatives, you are voting solely for your local members.
There are 76 Seats in the Senate, as opposed to the 150 in the House of Representatives. These 76 Seats are divided up by State and Territory. Each State has 12 Seats representing them, and each Territory has 2 Seats representing them.
When you fill out your Ballot Paper, you are voting to elect those members you want to represent you, in order of your preference. In a normal election, half of the Senate seats are usually up for grabs. Because this is a Double Dissolution election, all of them are. This means that 12 people on that list (or two if you are in a territory) will be elected - which is why the instructions are to number 1-6 Above the Line or 1-12+ below the line, in order of your preference.
The AEC even developed a 'practice voting' tool to further explain how to vote.