Well, you had to ask.
To fully grasp the concept, one needs to have an understanding of basic electrical principles.
Typically, the water hose analogy is used to demonstrate the basic principles.
The supply water plumbing of a house is thought of as a battery. The hydrant valve would be a switch. The garden hose would be the conductor (battery cables) carrying water (current) to a sprinkler head (starter).
The standing water pressure of the house plumbing (in PSI) is analogous to "battery power" (measured as voltage).
In this comparison, opening the hydrant valve allows water to flow through the hose to the sprinkler, causing it to spin.
In an automotive setting, operating the solenoid allows electrical current to flow through the battery cables to the starter, causing it to spin.
Various factors affect the performance of an electrical circuit much the same as variables in plumbing will affect the spinning sprinkler.
If you reduce the PSI of the house water supply, you expect the sprinkler to spin more slowly, similarly a reduction of battery voltage will result in a slower spinning starter.
If you change from a 3/4" diameter hose to a 3/8" diameter hose, the sprinkler will spin slower. If you change from the OEM battery cable size to some wires half their size, the battery will spin slower.
If you change the sprinkler out for one that is hard to turn, it will spin slower than the sprinkler that is free to spin. The same goes for a starter which is sticky or binding.
The analogies described are very similar except that in electrical systems you will not be able to make changes or affect the components without suffering some serious consequences (to the components).

The term "Amperes" is a tag used in describing the electrical flow (current) of an electrical circuit..
A battery does not "deliver" amps to the starter as though it were offering something "unsolicited". It is more like the starter is DEMANDING the battery give it the power it needs to turn the engine. So the starter is really at the mercy of the battery as to what it receives.
Batteries are rated by their capability to furnish power described as "X number of Amps". But that rating is only true if the battery has a full charge and the battery is new.
The analogy between water pressure and voltage is pretty accurate.
In plumbing, pressure of a specific value will push a specific volume of water through a specific diameter conduit having a specific value of resistance to flow.
In electrical systems, a specific voltage value will push a specific volume of electrical current through specific size conductors where there is a specific electrical resistance.
The consequence to be faced in electrical systems where the "specifics" become "imbalanced on the low side" is heat production.
It requires adequate voltage to push sufficient amps (in response to current draw) through a conducting circuit resulting in normal operation of the circuit components as it was designed for.

Consequence of low voltage:
If you cut the voltage in half but leave the other factors the same...........the result will be heat production due to the voltage not being adequate to push those amps through the resisting component.

Consequence of insufficient amp capacity of battery:
This is what we face when we are dealing with an old battery (anything that is not brand new).
Amp capacity of a battery is relevant to its physical size (plate surface area) and the state or condition of the battery.
Imagine two 12 volt batteries. One of the batteries fits a yard tractor, the other fits a large diesel engine agriculture tractor.
Both have a voltage level of 12, but the amp capacities are vastly different between the two.
The starters of each of those tractors is matched to the amp capacity of the batteries (and vice versa).
Basically and fundamentally speaking, battery amp capacity is chosen for an application with the goal of keeping VOLTAGE DROP to a minimum when the starter load is connected to the battery.
If you connect the yard tractor starter to the yard tractor battery you can expect perhaps a 3 to 4 volt drop in battery voltage after the starter begins spinning the engine, and that is acceptable.
If.....you connect the yard tractor battery to the agriculture tractor battery, you might possibly observe a battery voltage drop of .3 to .4.
But...........if you connect the starter of the big tractor to the yard tractor battery you could realistically expect the battery voltage to drop to zero. You can also expect the yard tractor battery to explode because that kind of load would be the same as connecting a battery cable between the posts of the small battery........in essence "a dead short".
The explosion is the result of the voltage drop causing catastrophic heat build up and flash melting of the plates inside the battery case creating a steam explosion as the electrolyte vaporizes.............combined with ignition of the explosive battery gasses.

Now back to the job of trimming a lawn.
When a battery is "old", its amp capacity is less than when new, and so will not be able to meet the current draw requirements of the starter when asked to. This will cause the battery voltage to drop, further reducing the ability of the battery to supply the needed amps.
When a battery is under charged (low voltage level) there is insufficient "pressure" to push the required amps through the starter, so heat develops quickly throughout the circuit.
Subjecting the starter to a low voltage/high amp draw scenario on a regular basis will lead to cumulative damage to the starter as well as other links in the circuit (key switch, starter solenoid, cables/wires/connectors)
Always buy the largest amp capacity battery you can get that will fit in your "whatever" , keep it fully charged,........and you will have a happy life.