Translation:The cat and the dog are running onto the street.
Your translation is grammatical, but describes something different.
Sie laufen auf die Straße. They are running onto the street, like they were on a yawn before and now move onto the street.
Sie laufen auf der Straße. They are running on the street, like they are running from location1 to location2 and they use the street instead of the sidewalk.
Well, not exactly, rather: accusative is used as the target of a transition, while dative is used to identify the location where something is taking place, regardless of movement or change of state. That is to mean that both whether you are sitting in the garden or whether you are jumping into the pool in the garden, the garden is dative because the activity (stationary or dynamic that it may be) is taking place entirely within the garden and not implying movement away from or towards the garden from somewhere else.
Well, I can translate the examples I gave: ‘ich sitze im Garten’ (dat., answers ‘wo sitze ich?’ = ‘where am I sitting?’); ‘ich springe in den Pool’ (acc., answers ‘wohin springe ich?’ = ‘where do I jump to?’); ‘ich springe im Garten’ (dat., answers ‘wo springe ich?’ = ‘where am I while I'm jumping?’).
A different sentence using both accusative and dative with a motion verb could be: ‘Hans geht ins Bett [accusative, direction] in seinem Haus [dative, location]’ = ‘Hans goes to bed in his own house’.