How can I install a Leviton timer switch in this box?
We start by numbering the cables in your diagram from left to right:


is an always-hot feed backstabbed into the single pole switch (it's always hot because it's backstabbed at the same end of the switch as the incoming hot feed).
is another always-hot feed (were I to guess, this is the circuit coming in, but there's an off chance some dunderhead wired it to come in on cable #1 instead).  I can tell because of how it's double-looped at the switch terminal screw then pigtailed off to the rest of the switches.
is the outbound switched hot from the single pole switch to whatever lighting is on it.  (It is connected to the other end of the switch from the hot terminal.)
is the outbound dimmed hot to its corresponding lighting load.
and 6. are the curious cases -- you'd expect both travelers to be in 6. but perhaps the two cables are run together as part of the same circuit in a California 3-way configuration?  (The traveler terminals on a 3-way switch are both at the same end.)  Either way, they're irrelevant to your timer trouble.


From here, we can tell where your error is -- you wired the switched hot on the timer to cable #1's hot, when the original single pole switch was switching cable #3's hot.

So, turn off the power to the circuit, then unnut the hot (black) and switched  (red) wires on the timer, and unwind the switched wire from cable #1's hot and cable #3's hot from the pigtail junction.  Then use new wirenuts to nut cable #1's hot into the pigtail junction and cable #3's hot to the switched wire from the timer.  Put everything back together, turn it back on, and enjoy your new timer!