Buying hits is a method to speed up play when the result of the roll really won't effect game play. The rule for doing so is to divide the total die pool used in the check by 4 and round down, the total is the number of hits for that check. (The rule is found on page 45 of the main rule book) Some GM's will allow it's use more frequently to speed up play, others will require a roll on any check that could have a long lasting effect. (For instance I've sat at tables that allow buying hits for magic drain resistance, where as a GM I never allow this...you can always have a bad roll)
The references for buying hits during downtime is a missions mechanic. It is typically used whenever you need to make a check during downtime (there may be exceptions to this), the reason for this is that GM's may not be available to witness any or all downtime rolls. I find this most used for buying gear between missions, but it is not limited to this. For instance a more convoluted example would be...
Hermetic mage has Magic rating 6, Logic 5, Willpower 5, summoning 6 (Fire Spirits +2), Binding 6 (Fire Spirits +2), Spirit Binding Focus 2 (Fire)
Total Die pools of
Summoning: 12 (14 for Fire spirits)
Binding: 12 (16 for Fire spirits)
Drain resistance: 10
During downtime the mage decides to summon and bind a force 7 fire spirit.
summoning die pool for fire spirit is 14, buying hits = 14/4 round down. =3 hits
Spirit resists the summoning with it's force, 7 dice, buying hits = 7/4 round down. =1 hit
Drain damage for this summoning is = to twice the hits rolled by spirit, min of 2, in this case drain DV = 2
drain resistance by mage has a pool of 10, buying hits = 10/4 round down. =2 hits (No drain received for the summoning)
mage nets 2 hits and has 2 services, then uses the appropriate amount of reagents and attempts the binding
Binding die pool for fire spirits is 16, buying hits = 16/4. = 4 hits
Spirit resists the binding with twice it's force, 14 dice, buying hits = 14/4 round down. =3 hits
Drain damage for this binding is = to twice the hits rolled by spirit, min of 2, in this case drain DV = 6
drain resistance by mage has a pool of 10, buying hits = 10/4 round down. =2 hits... the mage takes 4 damage from the drain (physical damage too since the force of the spirit was higher than the mages magic attribute)
However even though the mage took damage, the mage had 1 net hit on the binding and successfully bound the force 7 spirit with 2 services
Hopefully this clears up your confusion of what buying hits means and how it works