The sentence in the OP is idiomatic with a very specific meaning and intention, which may be hard to fully explain, but I'd like to add the following to the otherwise excellent comments and explanations above,
Skulle is a fairly versatile modal verb used in commands (frequently with emphasis,
du skal være færdig i aften) or to indicate an obligation/encouragement (
du skulle (=burde) ringe til din bedstemor). It is also used (without emphasis) to form the simple future tense (
jeg skal til København i næste uge).
Nu skal du næsten lige se denne dette her også (referring to the apple tree, æbletræet)
In the example above we can argue whether it is a polite command of sorts, like a friendly suggestion, or if it just indicates that the action will take place in the (near) future. Regardless, the modal verb is significantly softened not just by "lige", i.e.
du skal lige komme herhen, but by "næsten lige," which makes the statement sound more encouraging/inviting than commanding, i.e.
du skal næsten lige høre denne her sang.
Lige in this context is an adverb, modifying the modal verb and implies, as raumar says, "something minimal, "only" and "quickly, swiftly before you do something else".
Lige is used with a plethora of nuances and meanings in Danish, frequently modifying modal verbs (e.g.
skulle, ville, måtte) to minimize their otherwise stronger, commanding qualities. Please see here for more examples:
lige,3 — Den Danske Ordbog
Næsten adds an element of uncertainty, almost as if the speaker were open to other suggestions including being turned down. In this example it modifies 'lige'.
To convey the same type of a subtly polite suggestion in English, I think what comes closest is,
[...] Nu skal du næsten lige se dette her også. [...]
[...]
Now you might also want to see this. This is our apple tree. [...]
This is, of course, different from 'you might as well see this', which sounds too indifferent to my ear.
Best,
Bic.