In regards to this story
A single MP in the Houses of Parliament can say the word “object” and prevent a bill being passed.
Why can one MP block proposed legislation like this? What is the parliamentary rule which allows it?
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A single MP can't block a law from being passed; however, he can block a law being passed without debate. The Bill is a Private Members Bill, and there is always very limited time available to debate them, so although formally Bills that are objected to are put on the list to be debated later, in practice that rarely happens (except for a few at the top of the ballot). The time to lay out a case for or against the Bill would be in the actual debate.
Christopher Chope is from the libertarian right of the Conservative party and is a scourge of well-meaning backbench legislation. His objection in this case was reported as being that it was wrong to create a new, imprisonable criminal offence without a debate.
The government broadly supports the Bill, and that being the case, they may adopt it and allocate government time for a debate. And one should point out that it isn't legal throughout the UK, being already criminalized in Scotland.
Sittings of the Commons have a "moment of interruption" (the time of which depends on the day of the week) after which no further business can be done unless there is unanimous approval by the House. From Standing Order 9(6):
After the business under consideration at the moment of interruption has been disposed of, no opposed business shall be taken, save as provided in Standing Order No. 15 (Exempted business).
Any opposition - even a single MP objecting - appears to be sufficient to block a motion.
The time for the moment of interrupt for Fridays is quite early - 2:30pm - as it's the day when MPs typically spend time in their constituencies. Also, Fridays are reserved for private members' bills. The order in which they are considered is determined by ballot towards the start of the parliamentary session.
In the instance mentioned in the question - 15 June 2018 - the Mental Health Units (Use of Force) Bill was top of the list to be debated, and used up all the time available. At 2:30pm, the Speaker then went through the rest of the list. Of the 21 bills remaining, the second reading of every single one was objected to, and so they were all postponed to later dates.
The BBC article linked to in the question suggests that some MPs may object on principle to bills having their second readings passed without debate; or bills which the Government objects to may be blocked by their whips. This doesn't normally make headlines, but in the case, the Voyeurism (Offences) Bill had widespread - and Government - support, hence the outrage.
On the one hand, a bill having an undebated second reading would still be scrutinised at its other stages, and then again in the House of Lords. On the other hand, most private members' bills (especially those low down in the list) don't have much chance of becoming law in the first place, so delaying their second reading may make little difference.
The best chance for this bill (and indeed, any private members' bill) is for the Government either to allocate some of its time in the schedule to ensure its passage through the house, or to incorporate it into another Government bill.