Bum solid on the saddle during hard efforts. When you’re out the saddle, you don’t look like you’re ‘smashing’ it with your hips going on over the place, keeping everything smooth and helping power transfer. Core strength helps to angle the hips back so you can ride in a more tucked over position without crunching your cojones. Helps to avoid a sore lower back in long rides (and in general). You also feel a much better connection between each contact point on the bike.
I focused a lot this year on core strength and it really has helped. I wouldn’t say you have to push ‘less’ hard, although you’ll pedal more efficiently and reduce pain around the body. In terms of weight I’d say it helps reduce weight on the handlebars as you can ‘hover’ off them a little more. I also feel like it gives me an extra weapon in a sprint when you really want to be using each muscle you can.
Watch Alberto Contador here for an idea of what it ‘looks’ like instead of feels (around 2:05 minutes in), putting out a huge effort, and able to look smooth and controlled, driving his effort from the core (warning, terrible music):