False Super Saiyan is a transformation Goku takes when he has the physical power to become a Super Saiyan but lacks the mental need to become one
In this form, Goku's irises and pupils are no longer visible, mostly in turn to the power and bleak rage. His body bulks up a little, his hair straightens like that of a Super Saiyan, his skin color also appears to have a yellow hue to it, but there is no considerable change in feature other than a red tint to his hair. He gains a yellowish surge of aura, somewhat but not quite resembling that of a normal Super Saiyan's. In turn, the form somewhat resembles a glorified Kaio-ken form, but there are notable differences.
When he transformed into the true Super Saiyan transformation during the fight against Frieza, Goku witnessed the murder of Krillin (as well as the fact that Krillin isn't likely to be revived with the Dragon Balls, as Shenron had already granted that wish and could not resurrect the same person twice), something that drove him over the edge. This ultimate pain was not fully present during the fight against Slug, which is most likely why Goku did not transform fully. The form of False Super Saiyan was shown to be very erratic, lasting for only as long as the aura is ignited, and using up much more energy to sustain than the full Super Saiyan.
During the battle against Lord Slug, Goku's strength began to quickly debilitate. Having no more options, Goku's anger surged, causing him to gain a massive power increase, enough to quickly overwhelm Slug. However, after a quick sequence of Goku thrashing Slug around, his power quickly faded. It is safe to say that in a nutshell, the False Super Saiyan transformation occurs when a Saiyan has the required physical power but lacks the mental need to transform into a Super Saiyan. It is apparent however that the form gives a large power increase, enough for Goku, who was being easily beaten by Slug, to quickly overwhelm him. The power increase is never stated nor looked into, but it would be suggested that it at the very least, multiplied the user's power by around 10, seeing as how even Goku's strength enhancing Kaio-ken techniques were futile against Slug. Since the form was only used in Movie 4 its canonicity is often debated.
Goku is the only one in the Dragon Ball universe to be known to ever reach this form as it was possible that h ewas half-way through to becoming a Super Saiyan or had the physical power to beocme one but lacked the mental need to do so.