Perhaps one of the most overlooked, yet vital, aspects of training, especially weight training, is training intensity. This is one of the areas that can be the most difficult to progress in, yet yields the greatest returns. It's also one of the most misunderstood aspects of weight training. It's not enough to just train each muscle--it's not even enough to train each muscle well--you have to train each muscle intensely. Harnessing and improving your training intensity is easily the most effective way to get consistent improvements, period.
Aside from the more nuanced (read: mental) aspects of training intensely, there's the super obvious aspect that's most commonly associated with the term: how much overload are you applying to the muscle? Overload is how much adaptation are you are asking your body for. Adaptation will only occur if you tell your body that adaptation is required; your body doesn't know you're in a gym, it thinks it's out in the wild, relying on its strength for survival still LOL. So when the body is told that its exertions are sufficient, i.e. with a set of 12 easy reps, the body has no reason to improve. My 8th grade chem teacher in Finland taught me a valuable lesson about both electrons and humans: they're both extremely lazy and always take the path of least resistance. If a muscle is doing its job well enough, i.e. those 12 easy reps, why improve unnecessarily when those calories can be saved for survival? Therefore, in order to force change in the body we must overload, or tell the body that it needs to improve in order to survive. We can do this by training to failure.
Failure can mean a few different things. Depending on what you are training for, failure can be high weight/low reps, high reps/low weight or something in between. The same principle applies to conditioning as well, although failure is a little looser here--failing would mean death soo I would consider failure to be based on discomfort level instead. If you can run 4 miles easily, you will never get to 6 miles--or 4 miles faster--if you keep running the same way (easily).
Generally in the bodybuilding community, high intensity training, or HIT, is classified as one all-out set per exercise in a range of 6-10 reps--Dorian Yates/Mike Mentzer style. If you are doing 3 sets of 12 reps, you are not using HIT principles because you are not training to failure and you are completely wasting your time. Basically, if you "get" the last rep on your last set, you need to add moar weight because that was just a warm-up--nice try.
Another aspect of intensity is the mental "pushing" that occurs in the gym--how many more reps can you do than you told yourself you were going to do when you first started your set? How far into the pain zone can you go? Training to failure is an entirely mental thing. Your body is far stronger than your will to subject yourself to pain and discomfort. If you're thinking "Training isn't painful..." then you need to stop training like a granny and sack up. If you find yourself questioning the space-time continuum during 40 rep sets of leg press, then maybe you're approaching a 6 on a scale of 1 to 10. Like all values, your intensity level must be constantly re-evaluated. The standard for yesterday does not apply to today--today is different. You are infinitely capable of pushing yourself farther--intensity ad infinitum!
So how do people motivate themselves to constantly be more intense? Like I said--re-evaluation of all values! People also usually experience a lot of improvement in the beginning, but as strength gains start diminishing over time, it becomes necessary to lift the same weight, but make it more difficult. One way to do this is to improve or alter form, which is yet another aspect of intensity. The mental finesse you put into each contraction is as important as the exertion itself--are you feeling each fiber contract and stretch, or are you spacing out and letting your body go through the motions?
When I say that increasing your intensity is the best way to "increase your bench," I mean all three areas continuously: increase your mind-muscle connection to hit failure, so you can increase your reps to hit failure, so you can increase your weight to hit failure --> hit failure --> hit failure --> hit failure --> HUGE.
See, I do know what I'm doing.