The phrase (costs on the indemnity basis) means that if there was a dispute and you become involved in court action, the court can assess the winner's costs either on the standard basis, or the indemnity basis.
If the court assesses them on the indemnity basis, the court simply gives the benefit of any doubt to the winner, on whether the costs were reasonably incurred and payable or not (in favour of the receiving party - which would be the winner). If costs are awarded on the standard basis it gives the benefit of doubt to the paying party.
Costs are usually awarded on the indemnity basis if the conduct of the losing party is very questionable - meaning the winning party is likely to receive more in the way of legal costs by order of the court.
The UK company will have referred to the UK's Civil Procedure Rules, specifically Rule 44 :
CPR rule 44.3
(1) Where the court is to assess the amount of costs (whether by summary or detailed assessment) it will assess those costs –
(a) on the standard basis; or
(b) on the indemnity basis,
but the court will not in either case allow costs which have been unreasonably incurred or are unreasonable in amount.
(Rule 44.5 sets out how the court decides the amount of costs payable under a contract.)
(2) Where the amount of costs is to be assessed on the standard basis, the court will –
(a) only allow costs which are proportionate to the matters in issue. Costs which are disproportionate in amount may be disallowed or reduced even if they were reasonably or necessarily incurred; and
(b) resolve any doubt which it may have as to whether costs were reasonably and proportionately incurred or were reasonable and proportionate in amount in favour of the paying party.
(Factors which the court may take into account are set out in rule 44.4.)
(3) Where the amount of costs is to be assessed on the indemnity basis, the court will resolve any doubt which it may have as to whether costs were reasonably incurred or were reasonable in amount in favour of the receiving party.