The self destruct function and multiplying out of control are closely related.

Each time your cells multiply you lose a peice off the end of your chromosomes (where the alpha subunit attaches to begin replication.) The non-coding DNA that gets depleted in this way is called a telomere (literally 'end bit' in Greek)

Your body knows this is happening and if the telomere is too short for safe replication the cell will usually undergo apoptosis. Your body will even start becoming more conservative about cell multiplication as the telomeres get shorter so cells that would usually be renewed are expected to hold out a bit longer. This is the cause of ageing.

As you said some cells are able to bypass this mechanism and continue multiplying, ie cancer. The body can also produce an enzyme (telomerase reverse transcriptase) that is able to regenerate telomeres. This enzyme is usually expressed up until the age of about 21 which allows your body to grow and mature. Once you hit this age your body will stop producing the enzyme which causes the telomeres to deplete and begin the ageing process.

So of course the first thing these cells do once they're 'off the grid' is turn the immortality mode back on. They start to produce TeRT which regenerates and maintains their telomere length so that they can grow just like you did when you were a kid.