Design life can be one of two different things, and they aren't interchangeable.
A reference to '100 years design life' might mean that it's designed for a '1-in-100-years' loading case (wind load, or tidal surge, or whatever). This is solely about a means of quantifying the magnitude of loading. It is actually nothing whatever to do with the durability of the structure, it's about the strength of the structure.
The question asks about a different matter - the durability, and specifically the durability of reinforced concrete. It is quantified normally by reference to past experience in the particular environment telling you what the critical deterioration mechanism is likely to be, and then either reference to a standard solution, or a calculation of the lifetime for that mechanism. The calculations are normally to some degree empirical.
For a 'standard' structure, with a 'normal' exposure condition, 'normal' concrete characteristics, and 'normal' design life requirements, there will be standard solutions in the relevant design code, that probably simply defines the quantity of cover that will satisfy the design life. What constitutes 'normal' will depend upon the jurisdiction of the design code - different blends of cement are available in different parts of the world, and what is 'normal' for a national design code in a wholly-temperate country won't be 'normal' in the tropics or polar regions.
For example, in a structure in the splash zone in the Arabian peninsula, frost attack is not going to be a problem, but physical salt attack (or salt weathering) will be. Frost attack is where water freezing in pores and cracks expands and breaks the concrete. Salt weathering is where salty water is wicked up and evaporates at such a rate that salt crystals grow within the pores and break the concrete.
If a designer strays outside what their local design rules consider 'normal', or if the environment is especially aggressive, or the durability requirements are unusually onerous, then a specific calculation will be required.
The most common failure of reinforced concrete is that metallic reinforcement starts to corrode. Steel in concrete does not corrode because the concrete is very high pH, and steel in a high-pH environment is 'passivated' and does not corrode. However, slowly over time carbon dioxide from the atmosphere diffuses into the concrete and neutralises it. If you know the characteristics of your concrete, you can predict how fast that happens (by reference to empirical experience).
What normally actually triggers the corrosion (at least in marine or other salty environments - eg road salt), however, is chloride attack, where chloride ions diffuse in from the surface. Once the concentration of the chloride ions at the surface of the bar reaches a critical value, corrosion will soon take hold. You can calculate this, if you assume a concentration of chloride at the surface (from empirical data), and know the characteristics of the concrete (either empirical data, or by testing how fast chloride ions diffuse through it, but beware that as the concrete ages, its characteristics change, and you need to allow for that), and know the critical threshold (from empirical data).
There's a handy free program that does this calculation for you called Life-365, and it comes out of an American Concrete Institute committee. It does the chloride diffusion calculation for you, draws graphs and stuff, and if you're in the USA it even has the empirical data you need built-in so you don't need to look up what the local conditions are. (I use the program, but am not associated with it otherwise). The manual to the program has more detailed discussion of the science behind it, but the best thing is you can just play with it and see what effect changing something has on the life.
If you do the calc and you don't get enough life, then either you can put the reinforcement deeper (so it takes longer for the chloride to diffuse to it), or you make the concrete more resistive to chloride diffusing through it, or you use bar that needs a higher threshold value of chloride (stainless, say), or you surface treat the bar, or the concrete, or you put in galvanic or electro-chemical systems, or corrosion inhibitors, or something else. Lots of this stuff comes back to empirical data - they've tested it, and have test data that shows it will prevent corrosion for n years if you put in x amount of whatever.
