-When you have two populations on either side of a steep range of mountains, the fastest route between them is not over the mountain peaks, but through the lowest pass.
-The lower the altitude of the mountain pass, the easier it will be to go through the mountain range.
-But people don’t generally settle on the pass. They live in the valleys.

The TS is not a stable molecule. It has partial bonds and partial charges. It has an extremely short lifetime. But if you lower the energy of the transition state, like lowering the altitude of a mountain pass, you will increase the rate at which the reaction proceeds.

What’s faster to climb, a 10,000 foot mountain or a 15,000 foot mountain?

Depends on where you start! If your base camp for mountain #1 is at sea level, and your base camp for mountain #2 is at 14,000 feet, then climbing mountain #2 is easier! (assuming they are of equal difficulty).

In chemistry terms, the difference between the height (energy) of the basecamp (reactants) and the summit (transition state) is referred to as the activation energy. The rate of a reaction is related to the activation energy through the Arrhenius equation k = A e (-Ea/RT)

Computational chemistry techniques, in particular density-functional theory (DFT) with a software package like Gaussian is very powerful. For many examples see the beautiful work of Prof. Kendall Houk at UCLA: https://www.chem.ucla.edu/houk/ . To take one example of many, Paul H.-Y. Cheong, Robert S. Paton, Sarah M. Bronner, G-Yoon J. Im, Neil K. Garg, and K. N. Houk: “Indolyne and Aryne Distortions and Nucleophilic Regioselectivities,” J. Am. Chem. Soc., 132, 1267-1269 (2010)

Use a program like Gaussian to model the transition state using density-functional theory.