What it means when a spell counters or dispels another spell is addressed both by a specific entry in the FAQ and in answers to this question. In short, if a spell says that it dispels another spell, the spell's normal effect doesn't occur and, instead, typically, the spell automatically dispels that opposing spell—no check needed, and SR does not apply. Usually even the efficacy of the spells involved is unimportant: for example, a level 1 caster that manages to cast a haste spell automatically dispels a level 20 caster's slow spell, and, similarly, a caster's daylight spell that's modified by the feat Heighten Spell to a 9th-level spell dispels automatically any 8th-level or lower spell that possesses the descriptor darkness, even were the modified daylight spell's caster to cast it at caster level 8 from a custom staff and the darkness spell were cast at caster level 15.
When a wearer takes a free action to activate the boots of speed, the wearer usually gains the effects of the haste spell, but this GM would rule that if the wearer of the boots of speed activates them while affected by the spell slow, the haste effect that's normally bestowed by the boots is negated and, instead, the slow spell effect is automatically dispelled on the wearer only. (Each creature affected by the spell slow needs to be freed from the slow spell's effect individually; that FAQ entry covers this eventuality specifically using as an example a haste spell dispelling a slow spell on some affected creatures.) This GM would rule that this costs 1 round of the boots' 10 rounds/day haste effect.
(It's possible that another GM may rule differently: a legacy item from Pathfinder's antecedent Dungeons and Dragons 3.5, the boots of speed inherited almost verbatim its text from its predecessor, therefore both texts say that, upon the boots' activation, the wearer can "act as though affected by a haste spell." A GM may rule that since the boots don't really cast a haste spell, the slow effect continues. In such a case, the GM must determine how the wearer of activated boots is affected by the slow effect.)
If the GM agrees that the boots' haste-like effect dispels the slow effect, the wearer of the boots has so far only taken one free action this turn to activate the boots. Unless the GM enforces an arbitrary limit on the number of free actions a creature can take in a round—which, by the way, the GM totally can—, the wearer of the boots can on the same turn also take another free action to activate the boots a second time so that the wearer can actually benefit from the haste effect of the boots now that the slow effect has been dispelled by the boots' previous activation. In doing so, the wearer would exhaust a second round of the boots' 10 rounds per day.
With regard to the special ability the question quotes, assuming the GM agrees that a spell affecting you is a spell within 30 ft. of you (I mean, this GM would, I guess), then, upon the haste effect of the boots dispelling the slow spell, the wearer of the boots would heal 1 hp per 2 character levels. Likewise, were a creature that possesses that special ability to be within range of the boots' wearer when the slow effect is dispelled, that creature would benefit from that special ability similarly.
