Apart from the equations and calculations, a good way to see what is going on is to draw a spacetime diagram.
Here's the latest version of my spacetime diagrammer (time-upwards)
(version 4)
https://www.desmos.com/calculator/kq2qnojphq
(version 6)
https://www.desmos.com/calculator/sk2zhnmjmm [new]
I use a relative-velocity of (3/5)c above (
@Nugatory 's suggested velocity), but I invite to you to adjust the velocities.
The key idea is that "space is perpendicular to time",
a shortened slogan for
"an observer's spaceline is perpendicular to her timeline"
where perpendicularity is defined by the tangency to a "circle" centered at the tail of the displacement (at the origin).
The tangent [space]line of the
red observer meets the worldline ( [time]line ) of the
green observer
at the event marked by the
green cross at (x=0.6,t=1). [You may wish to turn on the grid using the Desmos-wrench in the upper-right corner.]
The displacement to the
green cross from the origin is 0.8 of the
green radial [time]line segment to the hyperbola. (You can measure this ratio of parallel segments using any ruler.)
As expected by symmetry,
the tangent [space]line of the
green observer meets the worldline ( [time]line ) of the
red observer at the event marked by the
red cross at (x=0,t=0.8).
The displacement to the
red cross from the origin is 0.8 of the
red radial [time]line segment to the hyperbola. (You can measure this ratio of parallel segments using any ruler.)
This ratio 0.8 is (4/5) = 1/(time dilation factor).
Red says the distant
green-cross event occurred 1 tick after the origin.
Green says the local
green-cross event occurred 0.8 ticks after the origin.
The time-dilation factor according to
Red is
\gamma_{red}=\cosh\theta = (adjacent to local)/(hypotenuse to distant) = (1 tick)/(0.8 ticks) = (5/4).
Green says the distant
red-cross event occurred 1 tick after the origin.
Red says the local
red-cross event occurred 0.8 ticks after the origin.
The time-dilation factor according to
Green is
\gamma_{green}=\cosh\theta = (adjacent to local)/(hypotenuse to distant) = (1 tick)/(0.8 ticks) = (5/4).
Adjust the v2-slider to see that this ratio is common to both observers.
(These are similar-triangles where similarity is defined by the "circle".)
Although I limit the range of the velocities to 0.98c,
you can type in 0.995 to override it. [You could also put "\tanh\left(\operatorname{arccosh}\left(10\right)\right) "]
(You might have to zoom out to see the tip of the green-tick.)
(To get more decimal places, open the "time dilation" folder and change the number of digits in round().)
Back to v1=0 and v2=0.6,
you can drag the event on the green worldline to the green-cross event
to determine the event on the red-worldline that green says is simultaneous with the green-cross event.
(To make the arithmetic simpler, you can drag the events controlling the spacelines to 5 ticks.)
You might wish to adjust the E-slider from "1" for Minkowski-spacetime
to "0" for Galilean-spacetime and "-1" for Euclidean space.
You can also adjust both v1 and v2.
Have fun.