In recent months we have had stories, mostly from college campuses, about speaking events being shut down by organized violence, pulling fire alarms and so forth.
As I see it there are two outcomes that can result from my opponents being given a platform to speak:
A) They will make fools of themselves by putting forth weak arguments
Or
B) They will make strong enough arguments for me to change my mind
So why should we fear people being given a platform to express their views?
There are four distinct reasons someone wants to shut down a speaking event:
Any sort of social event of a certain political bent is, net net, a win for that political side.
I have discussed this in a separate answer about the purpose of gay pride parades, there are tangible political benefits - from like minded people having an event. Whether it is a show of strength (hey, i'm not one of only two people like me around, powers that be!), to increasing own morale (hey, there's a ton of us!), to just socializing/networking. A speaking event adds on an informational component - the speaker presumably informs/teaches the audience something.
By shutting down a speaking event, you deny those whose politics you dislike that set of benefits.
As a reverse, it's an effective political tactic.
When opponents shut down your events, you get demoralized. Especially when they do it with clear "yah, fine" from The Man (in this case, campus administration).
Some people just feel better about imposing their will on others.
This was explored in the infamous Stanford prison experiment. You give someone power, and they enjoy exercising that power.
It creates a general chilling effect on one's political opponents.
If someone on one political side sees that their opponents can shut down a famous celebrity speaker using threats of violence, you can be sure they would shut up and not speak up on campus about their own political views, out of simple self preservation.
Additionally, specific to the topic at hand, the events are being shut down by radical progressives (mostly in USA).
Before delving to specific reasons why on their side, let's dispense with the expected propaganda claiming that "they are opposing nazis". This sort of "deplatforming" has very little to do with "nazis", "alt right", "Trump" or any other excuse. First, because 90%+ of those who are deplatformed aren't by any stretch of imaginatuion "nazis", even if they happen to hold right wing or conservative views (and, increasingly, left wing views in most topics). Second, by virtue of the fact that it started back in early 2000, when 'nazis' were meant as members of Germany's National Socialist German Workers' Party; "alt right" wasn't a thing and Trump was famous for casinos, not politics and was still a Democrat.
First African American US Secretary of State Condoleeza Rice was deplatformed at Rutgers in 2014
Left wing comedian Bill Maher was protested at Berkley in 2014
Richard Dawkins (formerly a darling of the left for radical atheist views, back when he only criticized Christians) was deplatformed, though not by college, in 2014.
FIRE's database of speech disinvitations shows a dramatic rise since 2000. Among them are:
G.W. Bush and Ari Fleischer (his press secretary). Ok, we can see that Republicans are not in vogue. Carly Fiorina. Rudolph Giuliani. on and on. Ben Carson.
Madeline Albright. Yep, Clinton's SecState, politically left. Thee were attempts to disinvite from both right AND left.
David Horowitz. Yet another noted neonazi, i'm sure.
Ben Shapiro (while pretty solid conservative, it's hard to accuse someone of being a neonazi when they got more antisemitic tweets than anyone else in entire year from actual neonazis).
Maryam Namazie in Europe (human rights Iranian campaigner)
Anti-female-genital-mutilation protestor Ayaan Hirsi Ali
So why is it that it's the progressives that is intent of not letting people who they disagree with speak?
It is due to the progressive's ideological inheritance, mostly Gramsci (with a further ago influence from Hegel).
I'll try to expand on this later, but a very good analyzis is in the 2004 book "Gramsci's Politics of Language: Engaging the Bakhtin Circle and the Frankfurt School" by Peter Ives; and its somewhat dense academic writing can be summed up thusly: language is materialistic (in Marxist sense, not physics sense), and therefore, speech and language informs and influences social reality, and therefore he who controls speech, advances towards controlling that reality.
This is in direct opposition to Milton/Locke tradition of free speech.
An somewhat different explanation is offered by left wing professors in the following form: deplatforming is "simply choosing the information that the students get to hear (or not to hear), as is the job of academy". Why it is that they get to decide based on their own political views - and not any sort of academic criteria - is not coherently explained in that article.
The context here is that people like Richard Spencer are not allowed to speak at all universities.
The reason some do not want them to speak is that they may be a (physical) threat to a certain part of the population, specifically to Black people, Jews, etc. While the speakers may or may not be violent themselves, Nazis and other far-right speakers attract similarly minded people who will be a threat.
Note also that your point B) is actually a fear. If you give Nazis and others of the far-right prominent opportunities to spread their ideology, it normalizes their ideas and they may convince some - even though their views are abhorrent - and give others already holding similar views the courage to be more open about it. And in the case of the far-right, this isn't about theoretical arguments or "having a different view"; history has shown what can happen if people who call for ethnic cleansing gain momentum.
As I see it there are two outcomes that can result from my opponents being given a platform to speak:
A) They will make fools of themselves by putting forth weak arguments
Or
B) They will make strong enough arguments for me to change my mind
You missed out C) - They will convince other people to join their cause. (or at least 'normalise' the position).
Furthermore, the argument doesn't have to be 'good' to do so. History is littered with cases of great orators whipping up popular opinion using nothing more than bombast.
In the cases cited by the questioner the silencing is a visceral expression of outrage and. most importantly, a bonding for disaffected youth craving power in an overwhelming and unresponsive world. It is, in short, a tantrum.
In other contexts, such as in state censorship of ideas, the other answers given here will be in play.
