Please note that the timeskip does not occur between Episode 1 and 2, but within Episode 2.
The full story is revealed in Season 3 Episode 6 and Season 3 Episode 11. Here is a recap of what happened, starting from a scene in Season 1 Episode 1:
Upon learning from Mikasa that Eren wants to join the Survey Corps,
Grisha is more obliging of the idea than Carla, asking his son why he
wants to leave the Walls. Upon hearing his son's answer, Grisha
announces that he will be departing, telling a bewildered Carla that
there is no point in trying to stop an inquisitive mind like Eren's.
Before he leaves, he promises to show Eren what he has been hiding in
their basement when he returns.
Later, during Wall Maria's fall,
Grisha travels to the Reiss Chapel in the night. Bursting in on the
royal family, Grisha [...] begs Frieda Reiss to use her Founding Titan ability to save
Shiganshina from the Titans that are breaking in, but she refuses. In
desperation, Grisha activates his own Titan ability, easily beating
Frieda in combat, and devouring her, before going on to slaughter the
entire Reiss family with the exception [of] Rod Reiss, who escapes.
Traveling to Trost District, Grisha locates Keith Shadis and enlists
his help in searching for his family among the refugees. Upon finding
Eren, the two men learn that Carla has been devoured, devastated by
the news. Telling Eren that he must avenge his mother, Grisha takes
his son into the woods. When Keith tries to stop him, arguing that
Grisha should not put such a burden on the boy's shoulders as he might
not be gifted like Grisha, Grisha simply explains that the fact that
Eren is his son makes him qualified, and orders Keith not to get
In the forest, he
entrusts Eren with the key to his Basement and injects him with Titan
serum, explaining to Eren that their memories will teach him how to
use the ability. Transforming into a Pure Titan, Eren devours Grisha
and returns to his human state, having gained the Founding Titan […].
And now, leading to Season 1 Episode 2, this is what happens:
Minutes pass when Keith unexpectedly hears thunder from the direction Grisha went with Eren, prompting him to venture into the forest to find them. He finds Eren lying unconscious with no sign of Grisha. He brings Eren back to the shelter to rest with Mikasa and Armin.
In Season 2 Episode 10, during
a conversation between Bertolt and Ymir, it is revealed that titan shifters cannot remember whom they have eaten prior to their transformation. Furthermore, having been a pure titan herself, Ymir claims that every day was like waking up into a nightmare. That is why Eren thought his transformation was just a bad dream, when in fact, it was not.
On top of that, Season 3 Episode 12 gives us a little insight on
the titan serum or titan injection itself. It is believed to be made up of titan spinal fluid. However, the exact chemical composition remains unknown.
So now we see that the timeskip encompasses
exactly one night.