Q:

How do Russia and the US know the other side keeps nuclear arsenals at agreed levels?

There are treaties between some countries that obligate the signatories to have not more nuclear weapons than specified in the treaty. How does the US know that Russia doesn't have more weapons than specified and vice versa?
What prevents any of the parties from silently producing more weapons than they are allowed to?
If a country puts tanks into a certain territory, you can see it with satellites. Bombers and ships are also big enough to be seen by the satellites. 
But if they manufacture more nuclear bombs than allowed and store them in a basement, how will the other party ever notice that they cheated? Nuclear warheads are smaller than tanks or warships, sometimes as small as a backpack.

A:

The New START Treaty signed by the United States and the Russian Federation in 2010 limited each side to 1,550 deployed strategic warheads.

Verification measures for New START are based on the 1991 START I
  Treaty and were modified for the purposes of the new Treaty. These
  measures include national technical means (e.g. satellites), on-site
  inspections and exhibitions, data exchanges and notifications related
  to strategic offensive arms and facilities covered by the Treaty, and
  provisions to facilitate the use of national technical means for
  treaty monitoring. To increase transparency and confidence, the Treaty
  also provides for the annual exchange of telemetry data on a parity
  basis, for up to five ICBM and SLBM launches per year.

Source: Nuclear Threat Initiative
The treaty provides for 18 on-site inspections per year.
Exclude a variety of nuances, both sides agree to let the other side remotely monitor or directly review their weapons manufacturing, testing, storage, and deployment capabilities to the extent required for verification. For example, building a "backpack" nuclear weapon still requires an industrial infrastructure and leaves a logistics trail; plus anyone wanting to rely on such a device would probably want to test it and there are systems in place to identify the seismic signatures of nuclear detonations.

A:

A Congressional Research Service paper "Monitoring and Verification in Arms Control" is a good summary on the topic.

Verification is the process that one country uses to assess whether another country is complying with an arms control agreement. To verify compliance, a country must determine whether the forces or activities of another country are within the bounds established by the limits and obligations in the agreement. A verifiable treaty contains an interlocking web of constraints and provisions designed to deter cheating, to make cheating more complicated and more expensive, or to make its detection more timely. In the past, the United States has deemed treaties to be effectively verifiable if it has confidence that it can detect militarily significant violations in time to respond and offset any threat that the violation may create for the United States.
The United States and Russia rely on their own national technical means of verification (NTM) to collect most of the information needed to verify compliance with arms control agreements. But, since the 1980s, the treaties have also mandated that the two sides share information through data exchanges and notifications, and conduct on-site inspections to confirm that information. The
verification regime in START used these monitoring measures not only to confirm that forces were consistent with the limits in the treaty, but also to detect and deter potential efforts to violate
the treaty.

The paper details various methods used:

National Technical Means of Verification (NTM)
These include things like satellite imagery, general spying, etc...
The paper provides an in-depth reference on NTM:

For descriptions of the systems included in the National Technical Means of Verification, see Scribner, Ralston, and Metz, The Verification Challenge, pp. 47-66. See also Ted Greenwood. Reconnaissance, Surveillance, and Arms Control. Adelphi Paper #88, London, International Institute for Strategic Studies, 1972;

Providing Telemetry Generated During Missile Flight Tests

On-Site Inspections

Voluntary Exhibitions

A:

What prevents any of the parties from silently producing more weapons
  than they are allowed to?

Technically nothing. That's why it silent If they can get away with it, then they get away with it. However both START treaties include provisions for both sides to do on-site inspections. The first START treaty included over 500 inspections (go down to Verification and Compliance). If there are signs of funny business going on, then the flag can be raised and the issue can be addressed. 
You also mentioned using satellites to spot tank and bombers and ships. Well it turns out the START treaties also make use of satellites. In addition to dismantling the bombs themselves, the START treaties have provisions for dismantling nuclear bomb delivery devices (aka bombers). Specifically both sides will bring their bombers to a giant open field and over a long period of time disassemble to bombers in a specific way out in the open so that it is visible with satellites. 

(random cool factoid: the planes were disassembled by dropping a giant steel blade from a crane to slice through the plane - essentially a giant guillotine)
On top of all that both the US and Russia both have huge intelligence programs. Building new nukes is hard and it takes a lot of industry and logistics. While its not impossible to hide, its certainly not easy.

