There was never an option offered of paying the outstanding fare and the train company’s costs. Train companies will sometimes offer that if they receive contrite responses which convince them that the behaviour is very unlikely to be repeated, but in the absence of that make use of the criminal law.
The train company told you, and the court, what financial loss they claimed you had caused them by a) depriving them of the revenue due for the journey you took and b) causing them to have to spend time investigating and processing the case against you. This was a) £4.10 and b) £150.
By pleading guilty to an offence you were then liable to be sentenced by the magistrates’ court, according to the statutory penalty for the crime (which is why the bundle of forms probably included space to indicate your income, since fines are partly income-based, along with space to submit any mitigation, that is argument that you should be punished less severely - via a smaller fine than usual. In railway cases there’s little room for that, and probably none in this case). The magistrate decided the punishment should be a fine of £333.
Finally, Parliament has decided that most people convicted of crimes should help to fund victim support services (since someone has to, and putting some of the burden on those who commit offences makes sense to the government) so there’s a surcharge of £133 to contribute to that budget.