Normally distributed random variables

Im finding it hard to understand why you can do the following transformation of to get from a normally distributed random variable with mean 0 and standard deviation of 1 to a normally distributed random variable with a specific mean and standard deviation:

mean + standard deviation

Finding it difficult to see the intuition behind multiplying by std. dev. and adding the mean.

This is a pretty fundamental thing to understand so would be grateful for any help.

Cheers,

Peter