In meinem Browser sind so viele Tabs offen, dass ich den Überblick verliere.

Here in describes location, not movement, so it takes the dative case.
Browser is masculine: der Browser. In the dative, mein becomes meinem, so:

• der Browser → in meinem Browser

If you were talking about motion into something, German would usually use the accusative instead.

Both are possible. German main clauses follow the verb-second rule, which means the finite verb has to be in the second position.

In this sentence, In meinem Browser takes the first position, so sind must come next:

• In meinem Browser | sind | so viele Tabs offen ...

If you start with the subject instead, that is also correct:

• So viele Tabs sind in meinem Browser offen, dass ...

The version in the sentence puts extra focus on where the problem is: in my browser.

Because so viele ... dass is the standard pattern for so many ... that.

• so viele Tabs ..., dass ... = so many tabs ..., that ...

By contrast, sehr viele just means very many / a lot of and does not naturally set up a result clause in the same way.

So:

• Ich habe sehr viele Tabs offen. = I have a lot of tabs open.
• Ich habe so viele Tabs offen, dass ich den Überblick verliere. = I have so many tabs open that I lose track.

Because dass here is a conjunction meaning that. It introduces a subordinate clause.

• ..., dass ich den Überblick verliere.

das with one s is usually an article or pronoun, such as the / that / which:

• das Buch
• Das ist gut.

A useful shortcut:

• If it introduces a whole clause, it is often dass
• If it points to a noun or stands in for one, it is often das

All German nouns are capitalized, so Browser, Tabs, and Überblick all begin with capital letters.

The singular is:

• der Tab

The plural is:

• die Tabs

It is an English loanword, and German uses it very commonly for browser tabs.

offen is a predicative adjective here, used with sein:

• Die Tabs sind offen. = The tabs are open.

Predicative adjectives in German do not take adjective endings. Compare:

• die offenen Tabs = the open tabs
• die Tabs sind offen = the tabs are open

So offen has no ending because it is not directly in front of a noun.

German often describes the state of something with sein + adjective:

• Die Tabs sind offen.

That means the tabs are in an open state. English often prefers I have tabs open, but German can express this either way.

You can also say:

• Ich habe so viele Tabs offen, dass ich den Überblick verliere.

That is also very natural. The original sentence just chooses to make Tabs the subject instead.

After dass, German uses subordinate clause word order, which sends the finite verb to the end.

So:

• Main clause: Ich verliere den Überblick.
• Subordinate clause: ..., dass ich den Überblick verliere.

That final verliere is completely normal and is one of the most important German word-order patterns to learn.

Because den Überblick verlieren uses Überblick as a direct object, so it must be in the accusative case.

• nominative: der Überblick
• accusative: den Überblick

The verb verlieren takes an accusative object, just like English to lose takes a direct object.

It is mostly an idiomatic expression. It means:

• to lose track
• to lose the overview
• to stop being able to keep everything straight

In this sentence, it means the speaker has so many open tabs that they cannot mentally manage them anymore.

Because in German, a subordinate clause introduced by dass must be separated by a comma.

So the comma in:

• ..., dass ich den Überblick verliere

is required. This is standard German punctuation, just like with other subordinating conjunctions such as weil, obwohl, and wenn.

Yes, but only if you do not need my.

• im Browser = in dem Browser = in the browser
• in meinem Browser = in my browser

You cannot contract in meinem the way you can contract in dem to im. So with my, the full form in meinem Browser is the correct one.