So, the main part of that sentence is the first part:
The service life for the xxx system shall be more than twenty years
That part I will assume is clear.
There's then another clause:
with the need for refurbishment running not less than fifteen years into this period
This has the subject "the need for refurbishment", the verb running, and the argument "not less than fifteen years". This suggests that there is some necessity for refurbishment, and that it will apply for at least fifteen years. The "into this period" determines that the fifteen-plus years will start at the same time as the twenty years' service life. That is to say, there will be a need for refurbishment for at least the first 15 years, and potentially longer.
Now, I would look at that and say "what?", because I don't know what sort of refurbishment they are talking about. That's what the overall meaning depends on. It might be that, elsewhere in the requirements, there is stated to be a need for refurbishment to keep the system up with changes in standards. That would mean the system would have to last over 20 years, and it would have to be refurbished to keep up with standards for at least 15 years. There are other possibilities, but that is the one I would consider most likely - but if that is what it means, there would be other things in the document to make that clear(er).