This sounds like infection - probably minor local infection rather than meningitis etc - but I don't think will have been caused by dirty equipment. The burning process will be as good at killing bacteria as it is at killing the goats tissue. It may have been in part due to operator error eg burning for too long, but it may be just bad luck as I am sure this will be a relatively common complication. If you burn a hole in your scalp it will leave a portal for infection to enter and of course goats will tend to be in dirtier places than a human and more likely to get wound contamination.
The equivalent in humans would I suppose be to burn off a finger or toenail - burning of big toenail beds to treat ingrown toenail is done with a chemical however - and is not near to the brain. These do get infected sometimes.
Sounds sore but I don't think it is. Nevertheless I would not be too keen to have it done.
This from Liverpool university:
Disbudding / De-horning goats – Vet-Only procedure
The removal of the horn buds in goat kids, and/or the removal of horns from adult goats is a procedure which can only be carried out by a veterinary surgeon in the UK. The Royal College of Veterinary Surgeons considers the act of disbudding to be a mutilation, and as such requires serious thought before being undertaken, but it is accepted that for management and welfare reasons it is often necessary in larger goat enterprises.
As the horn bud grows extremely rapidly in goat kids they should be disbudded between 2-7days old, and must only be done by a vet with the kids under general anaesthesia +/- local nerve blocks. Goat kids have very thin skulls and disbudding carried out by an untrained operative can easily cause irreparable damage to the brain, either by direct heat transfer damage, or by facilitating the entry of bacteria into the skull leading to a potentially fatal meningitis.
It is perfectly normal for goat kids to be a little off colour for a few days after being disbudded, but they should be closely monitored for signs which may suggest complications, eg nervous signs such as a head tilt, vacant expression, fitting, or an inability to interact with their environments.
Adult horns can be removed, but it is a much more stressful event for the goat than removing the smaller horn buds early in life. Typically the goat would be anaesthetised and given local anaesthetic nerve blocks before having the horns removed by a length of cutting wire.
In all cases tetanus prophylaxis is recommended