Below is how I understand the sentence. Let's go step by step by considering each word as it comes, but keeping in mind the following word in the context.
The word "Nam" links to a previous statement and signals that what follows is a an explanation or fuller statement or implication of what was said before. We can translate it provisionally as "For" and think of it more colloquially as "for you see...."
Starting off a statement with "et" usually signals a coordination between two following things that will be separated by another "et" and which we would translate with "both...and...." The scope of the first "et" could be over the next noun or noun phrase, but since it is at the beginning, it could also be over the next statement or clause. In such cases, a translation into English might imbed "both" before a verb rather than come at the beginning of the clause as in Latin. We can understand the statement so far as "for you see both the following things are true:...."
The word "secundās" can mean several things by itself so we need the next word to narrow down its meaning, so let's consider "secundās rēs" as a whole. It likely means "favorable/fortunate things" and must be accusative feminine plural. We must await a likely verb to explain its accusative case.
The next word, "splendidiōrēs," means "more splendid/illustrious/shining." It is presumptively the same case, gender, and number as the previous words and should refer to the same entity. As a comparative, it signals that the thing it is compared with will follow, if it has not already been stated. It is also awkward semantically if immediately interpreted as part of the previous noun phrase, so its contextual semantics signal us to look later in the sentence for the phrase it does belong to. In other words, it is likely a secondary predication and a separate statement about "secundās rēs." Let's consider it with the next word.
When we get to "facit" ("makes/does"), we see that "splendidiōrēs facit" likely means "makes more splendid." We also now understand that the accusative case of "secundās rēs" makes it the object, giving us "makes favorable things more splendid." If we read "facit" with the following word, "amīcitia," we see that the latter is nominative singular and must be its subject, giving us "friendship makes favorable things more splendid."
The placement of "amicitia" after "facit" is unusual and so signals that "amicitia" is a known part of the discourse and is just an afterthought reminder. The implication is that the overall topic under consideration is what friendship does. Knowing this, we can resolve the scope of "et." The sentence so far is saying that friendship does both one thing and another not yet stated thing. We can have a provisional translation like: "Friendship both makes favorable things more splendid...."
The next "et" signals the second thing that friendship does or applies to. We have "adversās," which is the same case, gender, and number as "rēs" and so presumptively refers back to it, since no likelier word immediately follows. We can tentatively translate it as "adverse things" and presume that it is meant to contrast with "favorable things." We can also expect a repetition of the sense of "facit" or some similar verb to maintain the parallelism implied by the two instances of "et."
Now we get to "partiens" ("sharing") it is the same case, gender, and number as "amīcitia" and so presumptively refers back to it, since no likelier word immediately follows. It needs an object, which seems to be conveniently provided for by "adversās," apparently giving us "sharing adverse things." Now we encounter two problems.
First, "adversās" is a great semantic match for "secundās," but "partiens" does not match "facit." This warns us to look for another verb that will provide a match later in the sentence.
Second, the phrase "Sharing adverse things" by itself is normally interpreted in English as a gerund phrase, and not a participle phrase. The word "partiens" is a participle and cannot be a gerund. Since English cannot use a participle by itself in this way, we need a tentative translation that adds the noun back in, giving us: "friendship both makes favorable things more splendid, and friendship sharing adverse things...."
An alternative more correct understanding is to see that "partiens" is a dominant participle, like those commonly found in ablative absolutes. As an ablative absolute, it would be easy to see the participle as refer to a circumstance and mean something like "by/in sharing." Since it modifies something inside the clause (the understood word "rēs), it usually must match that thing in case, rather than be in the ablative. It nonetheless retains the meaning of predicating an additional circumstance instead of acting semantically as a simple attributive adjective. I will use this more complex understanding in the final translation, but leave it for now.
Next is "commūnicansque." The enclitic "-que" signals an "and" that applies to a natural or evident pair/group of things, giving us: "friendship both makes favorable things more splendid, and friendship sharing and participating in adverse things...." Again, "commūnicans" is really a dominant participle.
Last, we come to "leviōrēs" ("lighter"). It seems to have no verb associated with it but is parallel in structure, case, gender, and number to "splendidiōrēs" and giving us a semantic match for the prior "facit." Since Latin allows greater ellipsis than English we can presume that, mūtātīs mūtandīs, whatever semantically applies to "splendidiōrēs" applies to "leviōrēs," giving us "makes lighter" after another "facit" is added back in. This leaves us with another problem, however, since "makes lighter" needs an object and the available word, "adversās" is already taken by the participles in "partiens commūnicansque."
The solution is to realize that Latin allows different verbs to share the same object even in cases where English does not. If we restore the second object for the sake of translation, we get: "friendship both makes favorable things more splendid, and friendship sharing adverse things makes adverse things lighter." This "sharing" also relates to completing the parallelism and treating leviōrēs exactly like "splendidiōrēs" and pairing it with the same noun.
Now we can improve the style of the full translation by (1) trying to capture some of the compact ellipsis in the Latin, (2) bringing out the meaning of the dominant participles, (3) using the Latin participles in a way more idiomatic to English, and (4) being looser and more idiomatic with the vocabulary. One possibility is: "For you see friendship not only makes favorable matters more splendid, but also adverse ones more bearable by sharing and experiencing them together."
