Jusque is generally constructed with preposition "à" and means up to or until, ie both temporal and spacial notions:
• J'attendrai jusqu'à son retour. I'll wait until his return
• La réunion a duré jusqu'à huit heures. The meeting lasted until 8PM
• Il le conduit jusqu'à la frontière. He drives him (all the way) to the border
• Elle l'a raccompagné jusqu'à sa maison. She walked him/her up to his/her house
Note: if another preposition is used, "à" disappears: "jusque chez lui"
When "jusque" is followed by other adverbs: "alors, ici, où", it is elided:
• Jusqu'alors, il était calme Until then, he was calm
• La forêt s'étend jusqu'ici The woods extend up to here
• Jusqu'où êtes-vous prêt à investir ? Up to how much are you ready to invest?
• Ma pelouse est tondue jusque-là (note the hyphen, like other words constructed with -là: celui-là, ce jour-là...) My lawn was mown up to there
"à" is a preposition, you cannot skip it, otherwise the sentence has no proper meaning.
elision (drop a vowel and replace it by an apostrophe) is applied to the "jusque", not to the word coming after, since the objective of the elision is to ease pronunciation of "jusque" with "à"
In English, "to" is more general than "toward(s)." You can generally substitute the former for the latter, but not vice versa. "Towards" isn't strong enough to imply that she is the destination (which is what jusqu'à implies), since it could imply that he is simply walking in her direction and that she is in his path.