Varys was recruiting Dorne and the Reach to Daenarys' side.

There is a similar dialogue in A Feast for Crows between Prince Doran Martell and his daughter Arianne (who does not appear in the show):

[Arianne] narrowed her eyes. “What is our heart’s desire?” [Prince Doran]“Vengeance.” His voice was soft, as if he were afraid that someone might be listening. “Justice.” Prince Doran pressed the onyx dragon into her palm with his swollen, gouty fingers, and whispered, “Fire and blood.”

This conversation serves the same purpose as the show conversation, to announce that Dorne is finally coming into the conflict, and they are supporting the Targaryens.

Characters in universe would understand that Varys/Prince Doran are referring to the Targaryens, because "Fire and Blood" is their house words.

In the final scene, you are able to see a ship with the Dornish sigil sailing in the fleet. (Right side of the image)

We can assume that Varys' meeting in Dorne occurred well before the final scene showing the fleet leave Meereen.

Asynchronous storytelling is a common tool in Game of Thrones and explains most of the timing discrepancies that we see.

The creators of the show have already said that the timelines of different places is not in sync. Some places move forward in hours, some in days and some in months.

Writer Bryan Cogman, who wrote episodes 6 & 7 of season 6 had this to say

The timelines between the various storylines don’t necessarily line up within a given episode. For instance, the ‘Northern Tour’ Jon and Sansa embark on would probably take a couple weeks, but Arya’s storyline over the past few episodes only spans a few days. We realized a while ago that if we tied ourselves in knots trying to make all the ‘story days’ line up between all the characters the momentum would suffer.

Varys leaves Mereen in episode 8. He is seen in Dorne in episode 10. So, that's believable. He gathers the support of Dorne and Tyrell ships and goes back in the same episode to Mereen. Varys doesn't have teleportation power because you can see Tyrell and Dorne ships along with Greyjoy and Targaryian ships in the naval fleet of Dany. You can't teleport ships so it is to be assumed that Varys sent a word back to Tyrion telling him that Tyrell and Martels are behind Dany now, so wait up. Varys must have got back with a token ship of Tyrell and Martel each with some envoys to prove their support.

Here are the screenshots of Martells & Tyrells ships with sunspear & rose as the banner.

Prima della ribellione di Robert, i Tyrell e i Martels erano sostenitori della famiglia Targaryen. Le relazioni sono state rafforzate anche dal matrimonio. Quando Robert salì al trono, furono costretti a seguirlo come re poiché i Targaryen erano morti (secondo loro) e Robert aveva vinto il trono per diritto di conquista. Inoltre all'epoca era piuttosto tosto con un grande esercito con lui (dai un'occhiata alla Ribellione di Greyjoy ).

Varys era quindi a Dorne per informare i leader delle due case di supporto Targaryen che c'era ancora un Targaryen vivo che poteva salire al trono. Così viene mostrato mentre dice "Fuoco e Sangue", che sono le parole di casa di Targaryens.

Per quanto riguarda il teletrasporto, è proprio così che è fatto lo spettacolo. Nella prima stagione, il convoglio di Robert impiega un mese per arrivare. Eppure vediamo la lettera di intimazione e il convoglio arrivare nello stesso episodio. Inoltre, Varys è affiancato da navi Martell e Tyrell insieme a Dany. Quindi puoi praticamente dire che sono vicino a Dorne. Più vicino di quanto sembri.

Even though this single episode, and things seems to be happening after few hours. Actually its few days between events.Single characters take lesser time to travel while on land rather than big armies and groups of people. However, by boats it is almost same time for big armies and single person.

There is no special power shown with Varys, apart from he has more connections so he gets onboarded to next ship quickly. You can see littlefinger also, travelling and reaching different location frequently.

As both characters, don't have any strings attached so they move anytime and don't need inform family and friends.