So the first mistake here is thinking that the American stock market crash by itself set off a chain of events that led directly and inexorably to a decade long worldwide depression. A brief review of stock market crashes will show that only the 29 crash was followed by a depression of this magnitude. It's not a closed question by any means, but any attempt at answering it needs to look at worldwide conditions in interwar period and not just focus on a single event.

We do not yet have our hands on the Grail by any means, but during the past fifteen years or so substantial progress toward the goal of understanding the Depression has been made. This progress has a number of sources, including improvements in our theoretical framework and painstaking historical analysis. To my mind, however, the most significant recent development has been a change in the focus of Depression research, from a traditional emphasis on events in the United States to a more comparative approach that examines the experiences of many countries simultaneously.

-Bernanke

A very strong hypothesis currently is that the depression was caused primarily by a massive decrease in world wide money supply. This triggered deflationary financial crises and put people out of work as price / wage levels failed to adjust to the sharp drop in money supply. This occurred due to conditions unique to the interwar world economy. (As opposed to the earlier hypotheses popular during the 60's which simply blamed contractionary Fed policy and didn't consider the operation of the international gold standard.)

The international gold standard was reconstituted after a period of suspension during WW1. It was expected that it would operate reasonably smoothly as it did before (a depression of the magnitude of the 30's did not occur under the pre-war gold standard), but conditions had changed.

To be sure some of the changes were deliberate. The US Fed chose to contract money supply starting in 1928 (as shown in a drop in the money multiplier) which counteracted a significant net inflow of gold. However there was a basic deflationary trend wherein economic growth outstripped gold supplies (see Eichengreen). Beyond that a wave of banking crises struck the developed world starting in 1931 and central banks reacted to this through more contractionary monetary policy. Each country attempted to hoard gold, to increase the ratio of gold backing actual money supply.

This differed from the operation of the pre-war gold standard where countries coordinated with one another to prop up any one that suffered banking panics. On the stability of the dollar in the 1890's:

The stability of the prewar gold standard was instead the result of two very different factors: credibility and cooperation. Credibility is the confidence invested by the public in the government's commitment to a policy. The credibility of the gold standard derived from the priority attached by governments to the maintenance of balance‐of‐payments equilibrium...

First, the stability of the dollar as much as the stability of sterling hinged on international cooperation. The efforts of the Belmont‐Morgan syndicate were successful only because European countries were willing to part with the gold required by the United States. Second, international cooperation could be arranged in the absence of a central bank, but only through highly public and potentially embarrassing channels.

-Eichengreen

After WW1, these conditions changed. Within each individual country, the usual debtors (the working class) gained more power at the expense of the usual creditors (owners). Wage levels and unemployment became hot political issues to a degree unseen before. The early interwar years were characterized by explosive inflation in a number of countries which lasted into the mid 20's. Reforms were then introduced to insulate monetary policy from public pressure, but this also had the perverse effect of reducing central bank autonomy, and damaging the cooperative gold standard. Political disputes (such as over war reparations) and divergent theories of policy further contributed to the breakdown of cooperation.

With the erosion of credibility, international cooperation became even more important than before the war. Yet the requisite level of cooperation was not forth‐coming. Three obstacles blocked the way: domestic political constraints, international political disputes, and incompatible conceptual frameworks. Domestic interest groups with the most to lose were able to stave off adjustments in economic policy that would have facilitated international cooperation. The international dispute over war debts and reparations hung like a dark cloud over all international negotiations, contaminating efforts to redesign and manage the gold standard system cooperatively. The competing conceptual frameworks employed in different countries prevented policymakers from reaching a common understanding of their economic problem, much less from agreeing on a solution.

-Eichengreen

Okay, but you say, money supply according to basic theory has no effect. Cutting the money supply in half should just result in all prices and wages getting slashed in half and everything goes on as normal. So the second piece of the puzzle is why nominal shocks should produce real shocks. One possible explanation is that long term financial instruments (a mortgage for example) are set in nominal terms and not usually indexed to price levels. A drop in money supply tends to hurt debtors as the nominal amounts they owe do not shift automatically. This triggers a wave of loan repayment by whatever means possible and decreases economic activity in real terms. The repayment of loans (possibly through illiquid collateral) can then hurt banks as well since nominal assets are now replaced by real assets (with now low price levels) and put the solvency of the bank itself in jeopardy, triggering further crises and creating something of a feedback loop.

This debt-deflation argument is somewhat novel, but a longstanding observation has been that real prices and wages do not adjust instantaneously to money supply. Just looking at raw data without getting into the specific mechanisms of price adjustment:

First, during the worldwide deflation of 1930 and 1931, nominal wages worldwide fell much less slowly than (wholesale) prices, leading to significant increases in the ratio of nominal wages to prices. Associated with this sharp increase in real wages were declines in employment and output.

-Bernanke

It took some time for countries to leave the gold standard for complex reasons unique to each country, but when it did happen it was eventually accompanied by recovery.

Okay so this is a huge rambling wall of text, but TLDR:

1. The pre war gold standard was relatively stable as nations cooperated with one another to maintain convertability.
2. WW1 happens.
3. For a variety of complex reasons including international disputes, divergent national experiences and theories of monetary policy, shifting powers of political interest groups, a shift in balance of trade in favor of certain countries (US), international cooperation broke down.
4. Policy and events triggered a general worldwide contraction of money supply.
5. For a variety of complex reasons, the financial system and pricing mechanisms did not adjust to this contraction, and therefore real output and employment collapsed.

This is a massive oversimplification of the literature on the subject, so here's some further reading:

The Macoreconomics of the Great Depression: A Comparative Approach, Bernanke

Golden Fetters: The Gold Standard and the Great Depression, Eichengreen

The Nation in Depression, Romer

Freedom From Fear: The American People in Depression and War, Kennedy