I'm surprised it needs an explanation but:
In this country parties tend to hold the major power in Westminster. In each constituency a number of candidates put themselves forward for election; major parties tend to contest all.
Usually a lot of the votes are split between the two parties who have been dominant in the recent past, some areas are more likely to elect one party, other areas another.
However if a major section of the public are disgusted regarding how their voice is been ignored many will either abstain, or more likely switch their vote to a candidate or party that has consistently shown they want what the public voted for and were denied; and even some of those who lost the referendum may well be swayed by considering the denial of democracy as a priority issue.
So, take a simple view of a marginal seat which changes from term to term. For the sake of argument, in that constituency, 50% of otherwise Labour voters are disillusioned and vote UKIP instead. 50% of otherwise Conservative voters are disillusioned and vote UKIP instead. UKIP keeps their vote and picks up from any other party who aren't fussed about the public being ignored, and it gets over twice the votes of the next nearest party, thus being elected.
This is then repeated fairly commonly throughout the UK with safe seats staying put but many others temporarily becoming UKIP. The UKIP party becomes the political force in Westminster that you think it naive to believe that they can be.