I main support, so this is the daily struggle for me.
Basically what I do is start talking from champ select. Don't fill the screen, but just establish yourself as a friendly person that has game knowledge. Compliment the first pick on ban choices, or on someone for making a shrewd choice. Voice your concern about your team comp ("Guys, we are kinda squishy" is something I have to say in most of my games...) and/or point out a flaw in theirs ("Taric and Cait have anti-synergy, she will stay all the way back and Taric is better the closer he is to you" or "He picked Trynd into Maokai, he will just split-push all day, while Mao will win us teamfights"). This establishes you as a person that has done the reading and has game knowledge. This predisposes the team to listen to you later.
I use the loading screen to think of suggestions for my team. When we're waiting for 1:55, I throw around some suggestions, like "hey midlaner, you might want to rush Zhonya's against Zed" or "I'll get spooky ghosts against rango"). This further cements what you did in champ select.
After that, I play the laning phase and mostly communicate with my ADC. If I'm not bot lane, I communicate more often with the team, keeping an eye out for opportunities (I take TP when I go top so I can roam), and basically I make myself known to the team that I'm watching out for them. Whenever I go top I pick something like Lissandra and build her tanky (instead of damage) so I can become a tanky CC monster that sets up plays for the others. If I go mid, I often pick something anti-assassin (like Lissandra or AP Nasus) and focus on becoming an AP tank with CC, leaving the damage to the rest of my team. A willingness to let the others get the glory is what gets them to trust you and follow your advice (which you solidify by complimenting them on good plays). It's then up to you to cement that with advice and calls.
One time, an Akali was wrecking our Vel'Koz so I advised him to rush Abyssal and then get Banshee's after Deathcap. When he did, he was able to kill Akali when she dove him, and I pointed out "See? She can't assassinate you now." I also went there when I noticed Akali was starting to kill him, and basically camped his lane (as a support) in order to prevent him from feeling helpless and going on tilt. I feel that if I hadn't jumped on the problem the second I noticed it, we might have lost the game to a fed Akali and a raging or fatalist Vel'Koz.
I see my job as a support, other than setting up plays, to be about managing the emotional balance of 4 strangers. I have to test the waters and do some cold reading to figure out how to appeal to them, but I generally find that being the emotional rock of the team can be invaluable in order to galvanise the team into a winning force.
Establishing yourself as a knowledgeable figure that sets up plays for others can get people to listen when you want them to do something. It gives you credibility when things go wrong and people start fighting and you have to throw some verbal ice at the situation to cool it down, or when you need to advise a teammate to change strategy. Maybe you need to tell your top laner to stop trying to 1v1 the Yasuo and instead set up a bait so that the entire team can kill him when he overconfidently split-pushes and then rotate mid to force a 4v5 and gain a tower out of that. This has a way higher chance of succeeding if you've done what I said earlier and you frame your requests as logical suggestions. It's also important to type what you want to say as if you were saying it out loud to a person. Adding touches like "guys, don't worry" before you state your plan to come back makes you appear friendlier. It's also important to phrase your suggestions as plans rather than in an accusatory fashion. Rather than being like "akali stop diving, they blow you up immediately", you can be like "hey Akali, I notice that they focus you a lot, why don't you hide and wait for them to blow their cooldowns on us before you jump in?"
It's just a matter of practice and diplomacy.