The duration of the journey of a time traveler is not equal to the separation in time between departure and arrival. It might take Marty 1 minute of his time in the DeLorean to travel 30 years into the past. So the duration of his journey does not equal the separation in time between departure and arrival.
It looks curved. If no time travel takes place, it is straight. (Why?)
He thinks it inadequate because it does not give us time travel as we ordinarily conceive it. Say it takes Marty 1 minute to travel back 30 years where he meets his parents. On the two-dimensional picture, he arrives back along only one dimension of time when his parents existed. But to arrive back at precisely the time his parents existed he would have to arrive at the same time along both dimensions. So on the 2D view, he does not arrive back at the 1955 that his parents knew and lived, so he does not travel in time as we ordinarily conceive it.
Lewis proposes that we distinguish between personal and external time. The personal time of a traveler is the time on his wristwatch (supposing it is perfectly accurate). External time is the separation in the four-dimensional manifold (block universe). So Marty's journey takes 1 minute of his person time but 30 years of external time.
Kat is born on 1 June, 1985. On her 30th birthday on 1 June, 2015 she travels back in time to give her newborn-self a stuffed lamb. These events take place at different personal times of hers, but are simultaneous in external time.
On 1 June, 2015 Kat travels instantaneously from her 30th birthday party on the beach of Thailand to the same place 30 years in the future. Since travel was simultaneous, the point of departure and arrival are her same personal time, but different external times, since one is 30 years in the future.
Lewis says that the paradox rests on an equivocation between different senses of ability. Relative to one set of facts (i.e. that Tim travels back in time, that nothing stands in his way of killing his grandfather, etc.), Tim is able to kill his grandfather. Yet relative to another set of facts (i.e. that his grandfather lived), Tim is not able to kill his grandfather. Since there is not a single sense under which Tim both is and isn't able to kill his grandfather, the paradox is dissolved.