Residential A/C compressors are generally single phase motors and all single phase motors do not naturally rotate on their own, they need something to START them rotating. There are numerous ways to do that, each with its own set of advantages and disadvantages. For A/C compressors, they use what's called a "Capacitor Start" method wherein a capacitor is put into the motor circuit for the first second or so in order to get it to start rotating, then once it has begun rotating in the correct direction it is taken out of the circuit again with a device called a "Potential Relay" (in most cases) that knows when the motor gets to speed.
A "Hard Start" kit is just a bigger capacitor for the motor and new a Potential Relay suitable for that bigger capacitor. The working principle behind his recommendation is that your existing motor starting capacitor was sized for when the motor was new and fresh, now the bearings are wearing and there is more friction, so that original capacitor is not strong enough to make the motor begin rotating. Some people also seem to believe that capacitors get "weak" over time (they don't; they either work or they fail, there is no "weakening").
There is some debate as to the validity of that concept though, because motor people know that over sizing a capacitor does not make up for bad mechanics and in fact comes with a new set of problems, otherwise the motor mfr would have just used a larger capacitor to begin with. The original purpose of a "Hard Start" kit was actually intended to deal with situations where your line voltage is too low for the motor to start, so the larger capacitor gave it a boost. Sometimes it can be true that when your A/C was first installed, the line power was higher and now so many people in your area have added A/C units and/or other big loads that it is causing a voltage drop that you didn't used to have. So my theory is that because adding a Hard Start kit will fix that problem sometimes, some HVAC technicians have taken to looking at them as the "cure-all" for any sort of problems. But if your compressor is truly wearing out, a Hard Start kit will, at best, only prolong the inevitable. It might get you through the summer, but be prepared for having to have the compressor replaced sooner than later.
Unless you can determine that your line voltage is indeed lower than normal. Your utility or an electrician can tell you that.