This is several questions wrapped together:
How can I diplomatically see the company's financial information?
How strong a claim does a stockholder or warrantholder have to see the company's financials?
What information do I need to know about the company financials before deciding to buy in?
I'll start with the easier second question (which is quasi implicit). Stockholders typically have inspection rights. For example, Delaware General Corporate Law § 220 gives stockholders the right to inspect and copy company financial information, subject to certain restrictions.
Check the laws and corporate code of your company's state of incorporation to find the specific inspection right. If it is an LLC or partnership, then the operating agreement usually controls and there may be no inspection rights. If you have no corporate stock, then of course you have no statutory inspection rights. My (admittedly incomplete) understanding is that warrantholders generally have no inspection rights unless somehow contracted for. So if you vest as a corporate stockholder, it'll be your right to see the financials—which may make even a small purchase valuable to you as a continuing employee with the right to see the financials. Until then, this is probably a courtesy and not their obligation.
The first question is not easy to answer, except to say that it's variable and highly personal for small companies. Some people interpret it as prying or accusatory, the implication being that the founders are either hiding something or that you need to examine really closely the mouth of their beautiful gift horse. Other people may be much cooler about the question, understanding that small companies are risky and you're being methodical. And in some smaller companies, they may believe giving you the expenses could make office life awkward.
If you approach it professionally, directly, and briefly (do not over-explain yourself) with the responsible accountant or HR person (if any), then I imagine it should not be a problem for them to give some information. Conversely, you may feel comfortable enough to review a high-level summary sheet with a founder, or to find some other way of tactfully reviewing the right information.
In any case, I would keep the request vague, simple, and direct, and see what information they show you. If your request is too specific, then you risk pushing them to show information A, which they refuse to do, but a vague request would've prompted them to show you information B. A too-specific request might get you information X when a vague request could have garnered XYZ. Vague requests are also less aggressive and may raise fewer objections.
The third question is difficult to say. My personal understanding is some perspective of how venture capitalists look at the investment opportunity (you didn't say how new this startup is or what series/stage they are on, so I'll try to stay vague).
The actual financials are less relevant for startups than they are for other investments because the situation will definitely change. Most venture capital firms like to look at the burn rate or amount of cash spent, usually at a monthly rate. A high burn rate relative to infusions of cash suggests the company is growing rapidly but may have a risk of toppling (i.e. failing before exit). Burn rate can change drastically during the early life of the startup. Of course burn rate needs the context of revenues and reserves (and latest valuation is helpful as a benchmark, but you may be able to calculate that from the restricted share offer made to you). High burn rate might not be bad, if the company is booming along towards a successful exit.
You might also want to look at some sort of business plan or info sheet, rather than financials alone. You want to gauge the size of the market (most startups like to claim 9- or 10-figure markets, so even a few percentage points of market share will hit revenue into the 8-figures). You'll also have to have a sense for the business plan and model and whether it's a good investment or a ridiculous rehash ("it's Twitter for dogs meets Match.com for Russian Orthodox singles!"). In other words, appraise it like an investor or VC and figure out whether it's a prospect for decent return. Typical things like competition, customer acquisition costs, manufacturing costs are relevant depending on the type of business activity.
Of course, I wouldn't ignore psychology (note that economists and finance people don't generally condone the following sort of emotional thinking). If you don't invest in the company and it goes big, you'll kick yourself. If it goes really big, other people will either assume you are rich or feel sad for you if you say you didn't get rich. If you invest but lose money, it may not be so painful as not investing and losing out the opportunity. So if you consider the emotional aspect of personal finance, it may be wise to invest at least a little, and hedge against "woulda-shoulda" syndrome. That's more like emotional advice than hard-nosed financial advice.
So much of the answer really depends on your particular circumstances. Obviously you have other considerations like whether you can afford the investment, which will be on you to decide. And of course, the § 83(b) election is almost always recommended in these situations (which seems to be what you are saying) to convert ordinary income into capital gain. You may also need cash to pay any up-front taxes on the § 83(b) equity, depending on your circumstances.