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If it says "+1 strength for the remainder of the run", I assume it means the strength is increased automatically for every subsequent ice I encounter during the current run? I ask because in another thread it's been said that you have to increase strength of a program for each ice encountered... Not sure what's right...
"Until the end of the Run" is a special condition on some specific Shaper icebreakers.
Normally ICE loses any strength boost when the encounter is over. "Until the end of the Run" overrides this normal situation. The icebreaker *maintains* any strength boosts applied to it until the Run is over.
For example, if you spend 2 [credit] to boost Battering Ram from strength 3 to strength 5 to get through a Wall of Thorns, Battering Ram 'keeps' its strength boosts (not its break credits) until the Run is over. A second Wall of Thorns would require no additional strength boost, and if you encountered an un-advanced Hadrian's Wall, you would only need 2 more [credit] to get to strength 7 on your Battering Ram (and would then pay to break).
Once the run is over, these 'maintained' boosts are lost.
Try to keep in mind the golden rule (the card is always right and always overrides the rules). Answers to questions are almost always going to be the general or rules-based case because covering every exception is getting increasingly difficult as the card pool expands.